| L(s) = 1 | − 6·3-s − 10·5-s + 6·7-s + 17·9-s + 20·11-s + 216·13-s + 60·15-s − 180·17-s − 64·19-s − 36·21-s + 22·23-s + 25·25-s + 234·27-s + 956·29-s − 540·31-s − 120·33-s − 60·35-s + 176·37-s − 1.29e3·39-s + 92·41-s + 508·43-s − 170·45-s + 508·47-s + 491·49-s + 1.08e3·51-s + 344·53-s − 200·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.323·7-s + 0.629·9-s + 0.548·11-s + 4.60·13-s + 1.03·15-s − 2.56·17-s − 0.772·19-s − 0.374·21-s + 0.199·23-s + 1/5·25-s + 1.66·27-s + 6.12·29-s − 3.12·31-s − 0.633·33-s − 0.289·35-s + 0.782·37-s − 5.32·39-s + 0.350·41-s + 1.80·43-s − 0.563·45-s + 1.57·47-s + 1.43·49-s + 2.96·51-s + 0.891·53-s − 0.490·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.241161818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.241161818\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 6 T - 65 p T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 p T + 19 T^{2} - 74 p T^{3} - 1412 T^{4} - 74 p^{4} T^{5} + 19 p^{6} T^{6} + 2 p^{10} T^{7} + p^{12} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T - 18 p^{2} T^{2} + 1680 T^{3} + 4342123 T^{4} + 1680 p^{3} T^{5} - 18 p^{8} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 108 T + 7126 T^{2} - 108 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 180 T + 16130 T^{2} + 1159920 T^{3} + 81394131 T^{4} + 1159920 p^{3} T^{5} + 16130 p^{6} T^{6} + 180 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 64 T - 10462 T^{2} + 53760 T^{3} + 136795019 T^{4} + 53760 p^{3} T^{5} - 10462 p^{6} T^{6} + 64 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 22 T - 10677 T^{2} + 289806 T^{3} - 29356796 T^{4} + 289806 p^{3} T^{5} - 10677 p^{6} T^{6} - 22 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 478 T + 102955 T^{2} - 478 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 540 T + 162062 T^{2} + 37776240 T^{3} + 7205534163 T^{4} + 37776240 p^{3} T^{5} + 162062 p^{6} T^{6} + 540 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 176 T - 1394 p T^{2} + 3300352 T^{3} + 2680409179 T^{4} + 3300352 p^{3} T^{5} - 1394 p^{7} T^{6} - 176 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 46 T - 16373 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 254 T + 164793 T^{2} - 254 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 508 T + 21966 T^{2} - 14453616 T^{3} + 18170071603 T^{4} - 14453616 p^{3} T^{5} + 21966 p^{6} T^{6} - 508 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 344 T - 111666 T^{2} + 23306688 T^{3} + 13119050203 T^{4} + 23306688 p^{3} T^{5} - 111666 p^{6} T^{6} - 344 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 184 T - 332190 T^{2} + 8227008 T^{3} + 84855829051 T^{4} + 8227008 p^{3} T^{5} - 332190 p^{6} T^{6} - 184 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 94 T + 69521 T^{2} + 48376818 T^{3} - 49843309252 T^{4} + 48376818 p^{3} T^{5} + 69521 p^{6} T^{6} - 94 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 710 T - 52285 T^{2} + 32050110 T^{3} + 67491257756 T^{4} + 32050110 p^{3} T^{5} - 52285 p^{6} T^{6} - 710 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 1368 T + 1049542 T^{2} + 1368 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 76 T - 2974 p T^{2} + 42191856 T^{3} - 103906585597 T^{4} + 42191856 p^{3} T^{5} - 2974 p^{7} T^{6} - 76 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 24 T - 8170 p T^{2} + 8161728 T^{3} + 173952306051 T^{4} + 8161728 p^{3} T^{5} - 8170 p^{7} T^{6} - 24 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 142 T + 334921 T^{2} - 142 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 602 T - 872439 T^{2} - 105407190 T^{3} + 772372358212 T^{4} - 105407190 p^{3} T^{5} - 872439 p^{6} T^{6} + 602 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 1420 T + 2163846 T^{2} - 1420 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.16030290285399177716374615290, −6.96729764242020064324085844908, −6.78617461331917886766203928861, −6.45934967549676067311225191595, −6.39013306618711278354551506211, −6.24932790241625813223802780072, −5.89797077224770568771664157763, −5.76757902868227083269126685563, −5.57823961454964144793849939804, −5.02941317078660033027281035669, −4.59428081920767706840875841991, −4.54647977300185008613150342576, −4.31590212460130591880079807732, −4.21914598992614355880061856557, −3.83156128317909932177714398073, −3.52376277439931846131100380513, −3.38057337667824471040155723835, −2.66648371447013379622522788453, −2.61763525016264738760384322792, −2.27167260056783956316416666519, −1.55785439700661133349811633501, −1.20176975049359340562540958270, −1.00981958581558931194824487970, −0.74946005414749816707217841076, −0.44350181558914247001929677970,
0.44350181558914247001929677970, 0.74946005414749816707217841076, 1.00981958581558931194824487970, 1.20176975049359340562540958270, 1.55785439700661133349811633501, 2.27167260056783956316416666519, 2.61763525016264738760384322792, 2.66648371447013379622522788453, 3.38057337667824471040155723835, 3.52376277439931846131100380513, 3.83156128317909932177714398073, 4.21914598992614355880061856557, 4.31590212460130591880079807732, 4.54647977300185008613150342576, 4.59428081920767706840875841991, 5.02941317078660033027281035669, 5.57823961454964144793849939804, 5.76757902868227083269126685563, 5.89797077224770568771664157763, 6.24932790241625813223802780072, 6.39013306618711278354551506211, 6.45934967549676067311225191595, 6.78617461331917886766203928861, 6.96729764242020064324085844908, 7.16030290285399177716374615290
