| L(s) = 1 | + 4·2-s − 5·3-s + 12·4-s − 20·6-s + 32·8-s + 9·9-s − 5·11-s − 60·12-s − 95·13-s + 80·16-s + 25·17-s + 36·18-s − 120·19-s − 20·22-s − 124·23-s − 160·24-s − 380·26-s − 100·27-s + 213·29-s + 70·31-s + 192·32-s + 25·33-s + 100·34-s + 108·36-s + 134·37-s − 480·38-s + 475·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.962·3-s + 3/2·4-s − 1.36·6-s + 1.41·8-s + 1/3·9-s − 0.137·11-s − 1.44·12-s − 2.02·13-s + 5/4·16-s + 0.356·17-s + 0.471·18-s − 1.44·19-s − 0.193·22-s − 1.12·23-s − 1.36·24-s − 2.86·26-s − 0.712·27-s + 1.36·29-s + 0.405·31-s + 1.06·32-s + 0.131·33-s + 0.504·34-s + 1/2·36-s + 0.595·37-s − 2.04·38-s + 1.95·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.278265924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.278265924\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 142 p T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 95 T + 6252 T^{2} + 95 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 25 T + 9938 T^{2} - 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 120 T + 16610 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 124 T + 10478 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 213 T + 32464 T^{2} - 213 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 70 T + 60630 T^{2} - 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 134 T + 65970 T^{2} - 134 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 370 T + 108170 T^{2} - 370 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 82 T + 156270 T^{2} - 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 115 T - 93886 T^{2} + 115 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1024 T + 542198 T^{2} - 1024 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 760 T + 497810 T^{2} - 760 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 790 T + 417234 T^{2} + 790 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 850 T + 777726 T^{2} + 850 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 372 T + 307918 T^{2} - 372 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 190 T + 289866 T^{2} + 190 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 425 T + 897378 T^{2} - 425 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2110 T + 2255006 T^{2} + 2110 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 560 T + 355538 T^{2} - 560 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 675 T + 1619546 T^{2} + 675 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.647042146793991932716463188710, −8.341092884689061992656770906593, −7.72354721971584763170723912846, −7.53341217650319714311392979880, −7.11245900866147890743322980366, −6.76464096066078672624151942185, −6.16779200342518923377175609085, −6.11477352211567179245916388203, −5.48356057921225500575864264932, −5.42066584945684316182332664066, −4.75782630666610378149634752004, −4.40721626319383773412670299133, −4.22504494476864881523113716643, −3.78253898196096983012364303935, −2.79458973299934943379988520766, −2.78466959777961436920730858259, −2.18180651187915259221927870382, −1.76786936990960234916173605526, −0.852228279790076015817104523848, −0.33780274455178630759080385161,
0.33780274455178630759080385161, 0.852228279790076015817104523848, 1.76786936990960234916173605526, 2.18180651187915259221927870382, 2.78466959777961436920730858259, 2.79458973299934943379988520766, 3.78253898196096983012364303935, 4.22504494476864881523113716643, 4.40721626319383773412670299133, 4.75782630666610378149634752004, 5.42066584945684316182332664066, 5.48356057921225500575864264932, 6.11477352211567179245916388203, 6.16779200342518923377175609085, 6.76464096066078672624151942185, 7.11245900866147890743322980366, 7.53341217650319714311392979880, 7.72354721971584763170723912846, 8.341092884689061992656770906593, 8.647042146793991932716463188710
