| L(s) = 1 | + 2·2-s + 2·4-s + 5-s + 2·10-s − 11-s − 4·13-s − 4·16-s − 2·17-s + 2·20-s − 2·22-s + 23-s − 4·25-s − 8·26-s − 7·31-s − 8·32-s − 4·34-s + 3·37-s − 8·41-s − 6·43-s − 2·44-s + 2·46-s + 8·47-s − 8·50-s − 8·52-s + 6·53-s − 55-s + 5·59-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 0.632·10-s − 0.301·11-s − 1.10·13-s − 16-s − 0.485·17-s + 0.447·20-s − 0.426·22-s + 0.208·23-s − 4/5·25-s − 1.56·26-s − 1.25·31-s − 1.41·32-s − 0.685·34-s + 0.493·37-s − 1.24·41-s − 0.914·43-s − 0.301·44-s + 0.294·46-s + 1.16·47-s − 1.13·50-s − 1.10·52-s + 0.824·53-s − 0.134·55-s + 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−7.59708149153402218158879958117, −7.03477908276603020210779304212, −6.21101030419725238522849385481, −5.55294080899685406164716506758, −4.97685157209324421449824776256, −4.28111751945710160622264923994, −3.43654811333543968800336820680, −2.58331042621701060186581520596, −1.87316747898306017870215940641, 0,
1.87316747898306017870215940641, 2.58331042621701060186581520596, 3.43654811333543968800336820680, 4.28111751945710160622264923994, 4.97685157209324421449824776256, 5.55294080899685406164716506758, 6.21101030419725238522849385481, 7.03477908276603020210779304212, 7.59708149153402218158879958117
