| L(s) = 1 | − 3-s + 7-s + 9-s − 11-s − 3·13-s + 2·17-s − 3·19-s − 21-s + 4·23-s − 27-s − 9·29-s − 2·31-s + 33-s + 11·37-s + 3·39-s − 4·41-s + 4·43-s + 3·47-s + 49-s − 2·51-s + 4·53-s + 3·57-s − 3·59-s + 10·61-s + 63-s − 11·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.832·13-s + 0.485·17-s − 0.688·19-s − 0.218·21-s + 0.834·23-s − 0.192·27-s − 1.67·29-s − 0.359·31-s + 0.174·33-s + 1.80·37-s + 0.480·39-s − 0.624·41-s + 0.609·43-s + 0.437·47-s + 1/7·49-s − 0.280·51-s + 0.549·53-s + 0.397·57-s − 0.390·59-s + 1.28·61-s + 0.125·63-s − 1.34·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−15.67817730855008, −15.17562676999208, −14.63510250027890, −14.37981191271853, −13.36304372252398, −12.99400837173219, −12.58081474604352, −11.81981341502796, −11.42547110966719, −10.86763548389664, −10.33951614903428, −9.734346695982818, −9.175422919197291, −8.546072956840669, −7.724408108586839, −7.383135196084107, −6.796163544430280, −5.841153265318010, −5.612780941490216, −4.777165543120453, −4.329328000586251, −3.497334433846588, −2.614111659515147, −1.934370362108484, −0.9805931249973306, 0,
0.9805931249973306, 1.934370362108484, 2.614111659515147, 3.497334433846588, 4.329328000586251, 4.777165543120453, 5.612780941490216, 5.841153265318010, 6.796163544430280, 7.383135196084107, 7.724408108586839, 8.546072956840669, 9.175422919197291, 9.734346695982818, 10.33951614903428, 10.86763548389664, 11.42547110966719, 11.81981341502796, 12.58081474604352, 12.99400837173219, 13.36304372252398, 14.37981191271853, 14.63510250027890, 15.17562676999208, 15.67817730855008
