| L(s) = 1 | − 2-s − 4-s + 2·5-s + 4·7-s + 3·8-s − 2·10-s + 4·11-s + 13-s − 4·14-s − 16-s − 2·20-s − 4·22-s − 25-s − 26-s − 4·28-s − 10·29-s − 4·31-s − 5·32-s + 8·35-s + 2·37-s + 6·40-s + 6·41-s − 12·43-s − 4·44-s + 9·49-s + 50-s − 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.51·7-s + 1.06·8-s − 0.632·10-s + 1.20·11-s + 0.277·13-s − 1.06·14-s − 1/4·16-s − 0.447·20-s − 0.852·22-s − 1/5·25-s − 0.196·26-s − 0.755·28-s − 1.85·29-s − 0.718·31-s − 0.883·32-s + 1.35·35-s + 0.328·37-s + 0.948·40-s + 0.937·41-s − 1.82·43-s − 0.603·44-s + 9/7·49-s + 0.141·50-s − 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33813 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33813 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.93441868132861, −14.76647738807600, −14.04845283810508, −13.94772862957693, −13.15013456545629, −12.81650213657699, −11.89815628874318, −11.36365747440349, −11.04537080962114, −10.37247107783774, −9.781665111199090, −9.246257267401265, −8.939422620322604, −8.359222387520320, −7.658012349342836, −7.388877434190904, −6.440168701364685, −5.829166746385923, −5.260036695576759, −4.639001517494773, −4.081972392265009, −3.413798564485953, −2.150610364764650, −1.554213474926212, −1.313969994476919, 0,
1.313969994476919, 1.554213474926212, 2.150610364764650, 3.413798564485953, 4.081972392265009, 4.639001517494773, 5.260036695576759, 5.829166746385923, 6.440168701364685, 7.388877434190904, 7.658012349342836, 8.359222387520320, 8.939422620322604, 9.246257267401265, 9.781665111199090, 10.37247107783774, 11.04537080962114, 11.36365747440349, 11.89815628874318, 12.81650213657699, 13.15013456545629, 13.94772862957693, 14.04845283810508, 14.76647738807600, 14.93441868132861
