| L(s) = 1 | + 2·7-s + 4·11-s + 13-s − 4·17-s − 6·19-s + 4·29-s − 6·31-s + 2·37-s − 10·41-s + 8·43-s − 3·49-s − 4·53-s + 4·59-s + 2·61-s + 6·67-s − 8·71-s − 10·73-s + 8·77-s + 4·79-s + 12·83-s + 2·89-s + 2·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.20·11-s + 0.277·13-s − 0.970·17-s − 1.37·19-s + 0.742·29-s − 1.07·31-s + 0.328·37-s − 1.56·41-s + 1.21·43-s − 3/7·49-s − 0.549·53-s + 0.520·59-s + 0.256·61-s + 0.733·67-s − 0.949·71-s − 1.17·73-s + 0.911·77-s + 0.450·79-s + 1.31·83-s + 0.211·89-s + 0.209·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.80626530836655, −14.42822283933806, −13.91969282439151, −13.27433190038636, −12.86003473461190, −12.25199521919627, −11.61549537435902, −11.37316914886044, −10.62920312575078, −10.43925492903553, −9.419757276115243, −9.127255230308853, −8.472926226003134, −8.205475177101219, −7.364796649191534, −6.755213969981705, −6.383652455332150, −5.768455187026666, −4.948000347483416, −4.439627617115683, −3.964631932156950, −3.287952079380347, −2.293135469403591, −1.817581165578218, −1.074659495455045, 0,
1.074659495455045, 1.817581165578218, 2.293135469403591, 3.287952079380347, 3.964631932156950, 4.439627617115683, 4.948000347483416, 5.768455187026666, 6.383652455332150, 6.755213969981705, 7.364796649191534, 8.205475177101219, 8.472926226003134, 9.127255230308853, 9.419757276115243, 10.43925492903553, 10.62920312575078, 11.37316914886044, 11.61549537435902, 12.25199521919627, 12.86003473461190, 13.27433190038636, 13.91969282439151, 14.42822283933806, 14.80626530836655
