| L(s) = 1 | − 2·5-s − 2·13-s + 6·17-s − 4·23-s − 25-s + 2·29-s − 10·37-s + 6·41-s + 8·43-s + 4·47-s − 7·49-s + 6·53-s + 12·59-s − 2·61-s + 4·65-s + 4·67-s − 12·71-s + 14·73-s − 16·79-s − 12·83-s − 12·85-s − 10·89-s − 14·97-s − 6·101-s + 8·103-s + 4·107-s − 2·109-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.554·13-s + 1.45·17-s − 0.834·23-s − 1/5·25-s + 0.371·29-s − 1.64·37-s + 0.937·41-s + 1.21·43-s + 0.583·47-s − 49-s + 0.824·53-s + 1.56·59-s − 0.256·61-s + 0.496·65-s + 0.488·67-s − 1.42·71-s + 1.63·73-s − 1.80·79-s − 1.31·83-s − 1.30·85-s − 1.05·89-s − 1.42·97-s − 0.597·101-s + 0.788·103-s + 0.386·107-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−7.36161511466328377735501859289, −7.02675563347022380094744326266, −5.87782900052390037389579039459, −5.45948081850354739388901240840, −4.47796414004690219696123638365, −3.88497632421151025487572108929, −3.17526362527244961285126062896, −2.27028915996688623719547137958, −1.12848890972489228246073206230, 0,
1.12848890972489228246073206230, 2.27028915996688623719547137958, 3.17526362527244961285126062896, 3.88497632421151025487572108929, 4.47796414004690219696123638365, 5.45948081850354739388901240840, 5.87782900052390037389579039459, 7.02675563347022380094744326266, 7.36161511466328377735501859289
