| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 3·11-s + 15-s − 7·17-s + 3·19-s − 21-s + 23-s − 4·25-s + 27-s + 29-s + 8·31-s − 3·33-s − 35-s − 37-s + 4·41-s + 5·43-s + 45-s + 49-s − 7·51-s + 6·53-s − 3·55-s + 3·57-s − 10·59-s + 13·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.258·15-s − 1.69·17-s + 0.688·19-s − 0.218·21-s + 0.208·23-s − 4/5·25-s + 0.192·27-s + 0.185·29-s + 1.43·31-s − 0.522·33-s − 0.169·35-s − 0.164·37-s + 0.624·41-s + 0.762·43-s + 0.149·45-s + 1/7·49-s − 0.980·51-s + 0.824·53-s − 0.404·55-s + 0.397·57-s − 1.30·59-s + 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.516274007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.516274007\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.97967529722692, −12.70352125570585, −11.96833649399944, −11.63984900688741, −10.94102444877199, −10.58331147726277, −10.12947350825251, −9.543531401736316, −9.322174063004069, −8.673993562243344, −8.343622646239835, −7.632268954684260, −7.434684804862659, −6.561084353293001, −6.444133689334801, −5.705489767452454, −5.204833382753048, −4.622311457292040, −4.149426096797156, −3.548693454302911, −2.819289288082150, −2.473761161983984, −2.036969147243423, −1.171658454823654, −0.4294301661908647,
0.4294301661908647, 1.171658454823654, 2.036969147243423, 2.473761161983984, 2.819289288082150, 3.548693454302911, 4.149426096797156, 4.622311457292040, 5.204833382753048, 5.705489767452454, 6.444133689334801, 6.561084353293001, 7.434684804862659, 7.632268954684260, 8.343622646239835, 8.673993562243344, 9.322174063004069, 9.543531401736316, 10.12947350825251, 10.58331147726277, 10.94102444877199, 11.63984900688741, 11.96833649399944, 12.70352125570585, 12.97967529722692
