| L(s) = 1 | − 2·7-s + 2·13-s − 8·17-s + 6·19-s − 23-s − 5·25-s − 2·29-s − 4·31-s + 6·37-s − 10·41-s + 6·43-s − 3·49-s − 12·53-s − 4·59-s − 10·61-s − 6·67-s + 2·73-s − 6·79-s + 12·89-s − 4·91-s + 6·97-s − 6·101-s − 14·103-s − 14·109-s + 12·113-s + 16·119-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.554·13-s − 1.94·17-s + 1.37·19-s − 0.208·23-s − 25-s − 0.371·29-s − 0.718·31-s + 0.986·37-s − 1.56·41-s + 0.914·43-s − 3/7·49-s − 1.64·53-s − 0.520·59-s − 1.28·61-s − 0.733·67-s + 0.234·73-s − 0.675·79-s + 1.27·89-s − 0.419·91-s + 0.609·97-s − 0.597·101-s − 1.37·103-s − 1.34·109-s + 1.12·113-s + 1.46·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−9.237434102207363638201860341587, −8.171603902186240338008563091253, −7.35393631253367282172254155252, −6.48899758106257706190792833118, −5.87257923482558878497692549178, −4.77593113876989008025786234914, −3.83125738263606626167277658660, −2.93746481391339448160649201663, −1.71258596371455101275544465620, 0,
1.71258596371455101275544465620, 2.93746481391339448160649201663, 3.83125738263606626167277658660, 4.77593113876989008025786234914, 5.87257923482558878497692549178, 6.48899758106257706190792833118, 7.35393631253367282172254155252, 8.171603902186240338008563091253, 9.237434102207363638201860341587
