| L(s) = 1 | + 4-s − 8·5-s − 6·7-s − 6·9-s + 12·11-s − 12·13-s + 6·17-s + 6·19-s − 8·20-s + 38·25-s − 6·28-s − 4·31-s + 48·35-s − 6·36-s + 2·41-s + 12·44-s + 48·45-s + 8·49-s − 12·52-s + 36·53-s − 96·55-s − 12·59-s − 12·61-s + 36·63-s − 64-s + 96·65-s + 6·67-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.57·5-s − 2.26·7-s − 2·9-s + 3.61·11-s − 3.32·13-s + 1.45·17-s + 1.37·19-s − 1.78·20-s + 38/5·25-s − 1.13·28-s − 0.718·31-s + 8.11·35-s − 36-s + 0.312·41-s + 1.80·44-s + 7.15·45-s + 8/7·49-s − 1.66·52-s + 4.94·53-s − 12.9·55-s − 1.56·59-s − 1.53·61-s + 4.53·63-s − 1/8·64-s + 11.9·65-s + 0.733·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5175178898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5175178898\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 47 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 + 6 T + 4 p T^{2} + 96 T^{3} + 291 T^{4} + 96 p T^{5} + 4 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 12 T + 82 T^{2} + 408 T^{3} + 1611 T^{4} + 408 p T^{5} + 82 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 3 T + 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 6 T - 8 T^{2} - 36 T^{3} + 891 T^{4} - 36 p T^{5} - 8 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 55 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T - 12 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2 T - 31 T^{2} + 94 T^{3} - 620 T^{4} + 94 p T^{5} - 31 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 4266 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 8474 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 36 T + 642 T^{2} - 7560 T^{3} + 64187 T^{4} - 7560 p T^{5} + 642 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 2 T^{2} + 288 T^{3} + 8187 T^{4} + 288 p T^{5} + 2 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T - 11 T^{2} + 396 T^{3} + 11520 T^{4} + 396 p T^{5} - 11 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 100 T^{2} - 528 T^{3} + 4059 T^{4} - 528 p T^{5} + 100 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T + 32 T^{2} - 828 T^{3} - 7581 T^{4} - 828 p T^{5} + 32 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 100 T^{2} + 10086 T^{4} + 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 14 T + 64 T^{2} + 364 T^{3} - 4301 T^{4} + 364 p T^{5} + 64 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 12 T + 190 T^{2} + 1704 T^{3} + 17259 T^{4} + 1704 p T^{5} + 190 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 139 T^{2} - 18 T^{3} + 19500 T^{4} - 18 p T^{5} - 139 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 326 T^{2} + 44955 T^{4} - 326 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−8.308309749908091895568614838229, −7.86809572767764544746799004482, −7.77447218230762367604237294188, −7.32977481453183313854146117563, −7.21434555852360978173304763714, −7.07576806772942669766100840473, −6.90788807005502459156982043082, −6.85399168041481655328300829744, −6.26068618965867028918035359577, −5.99725054978614868554371862263, −5.79976790864534711134385056235, −5.52621835129458002959736948540, −4.93002327979365677985497953661, −4.84222687169843797407332435833, −4.45216539537335777350910946237, −4.03824625796922037485299798927, −3.83202000874439967995615623278, −3.60865885480631062549094738059, −3.36686209139285066436103425853, −3.03095238161201520402357817312, −2.85757778795856696267194885522, −2.62579009013886535276520687033, −1.68470661291711487312297580928, −0.67490343496289274759006744380, −0.49556059449336150711849857267,
0.49556059449336150711849857267, 0.67490343496289274759006744380, 1.68470661291711487312297580928, 2.62579009013886535276520687033, 2.85757778795856696267194885522, 3.03095238161201520402357817312, 3.36686209139285066436103425853, 3.60865885480631062549094738059, 3.83202000874439967995615623278, 4.03824625796922037485299798927, 4.45216539537335777350910946237, 4.84222687169843797407332435833, 4.93002327979365677985497953661, 5.52621835129458002959736948540, 5.79976790864534711134385056235, 5.99725054978614868554371862263, 6.26068618965867028918035359577, 6.85399168041481655328300829744, 6.90788807005502459156982043082, 7.07576806772942669766100840473, 7.21434555852360978173304763714, 7.32977481453183313854146117563, 7.77447218230762367604237294188, 7.86809572767764544746799004482, 8.308309749908091895568614838229
