| L(s) = 1 | − 4·7-s + 2·11-s − 13-s + 6·17-s − 4·19-s − 4·23-s − 5·25-s − 10·29-s − 8·31-s − 2·37-s − 4·43-s − 2·47-s + 9·49-s + 2·53-s − 10·59-s + 10·61-s + 8·67-s − 2·71-s − 10·73-s − 8·77-s + 8·79-s − 6·83-s + 12·89-s + 4·91-s − 2·97-s − 2·101-s − 16·103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.603·11-s − 0.277·13-s + 1.45·17-s − 0.917·19-s − 0.834·23-s − 25-s − 1.85·29-s − 1.43·31-s − 0.328·37-s − 0.609·43-s − 0.291·47-s + 9/7·49-s + 0.274·53-s − 1.30·59-s + 1.28·61-s + 0.977·67-s − 0.237·71-s − 1.17·73-s − 0.911·77-s + 0.900·79-s − 0.658·83-s + 1.27·89-s + 0.419·91-s − 0.203·97-s − 0.199·101-s − 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−9.681756717850981524698516360760, −9.030372930880778694181372835144, −7.86825833061979528151486382698, −7.06397869640228109190188534796, −6.14403274710687571852908041333, −5.50229892840847257590656724242, −3.93300941852702118688724477681, −3.38857153949511278983103885237, −1.93429528390897652987163936343, 0,
1.93429528390897652987163936343, 3.38857153949511278983103885237, 3.93300941852702118688724477681, 5.50229892840847257590656724242, 6.14403274710687571852908041333, 7.06397869640228109190188534796, 7.86825833061979528151486382698, 9.030372930880778694181372835144, 9.681756717850981524698516360760
