| L(s) = 1 | + 2-s − 4-s − 7-s − 3·8-s − 11-s + 13-s − 14-s − 16-s − 22-s − 2·23-s − 5·25-s + 26-s + 28-s − 4·29-s − 2·31-s + 5·32-s − 11·37-s + 7·41-s − 7·43-s + 44-s − 2·46-s − 6·49-s − 5·50-s − 52-s − 12·53-s + 3·56-s − 4·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s − 0.301·11-s + 0.277·13-s − 0.267·14-s − 1/4·16-s − 0.213·22-s − 0.417·23-s − 25-s + 0.196·26-s + 0.188·28-s − 0.742·29-s − 0.359·31-s + 0.883·32-s − 1.80·37-s + 1.09·41-s − 1.06·43-s + 0.150·44-s − 0.294·46-s − 6/7·49-s − 0.707·50-s − 0.138·52-s − 1.64·53-s + 0.400·56-s − 0.525·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 371943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371943 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5504703831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5504703831\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.64493194276501, −12.18900492327394, −11.70187192336317, −11.19272687276533, −10.81003589746465, −10.10622145351324, −9.812741898702011, −9.290863284282093, −8.996758240843418, −8.303807769700592, −8.000117640741724, −7.483237285821213, −6.815700137519338, −6.287407598233400, −6.025654054284021, −5.342027465107377, −5.039475082293442, −4.564413414990168, −3.780203531518661, −3.558654121891706, −3.210560020593113, −2.263260620778706, −1.950314681480659, −1.037424134436991, −0.1833129290934301,
0.1833129290934301, 1.037424134436991, 1.950314681480659, 2.263260620778706, 3.210560020593113, 3.558654121891706, 3.780203531518661, 4.564413414990168, 5.039475082293442, 5.342027465107377, 6.025654054284021, 6.287407598233400, 6.815700137519338, 7.483237285821213, 8.000117640741724, 8.303807769700592, 8.996758240843418, 9.290863284282093, 9.812741898702011, 10.10622145351324, 10.81003589746465, 11.19272687276533, 11.70187192336317, 12.18900492327394, 12.64493194276501
