| L(s) = 1 | + 3-s − 3·7-s + 9-s − 5·11-s − 13-s + 17-s − 5·19-s − 3·21-s − 2·23-s + 27-s − 9·29-s − 2·31-s − 5·33-s + 7·37-s − 39-s − 3·41-s + 2·43-s − 3·47-s + 2·49-s + 51-s − 11·53-s − 5·57-s + 10·59-s − 8·61-s − 3·63-s − 6·67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.242·17-s − 1.14·19-s − 0.654·21-s − 0.417·23-s + 0.192·27-s − 1.67·29-s − 0.359·31-s − 0.870·33-s + 1.15·37-s − 0.160·39-s − 0.468·41-s + 0.304·43-s − 0.437·47-s + 2/7·49-s + 0.140·51-s − 1.51·53-s − 0.662·57-s + 1.30·59-s − 1.02·61-s − 0.377·63-s − 0.733·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 6 T + p T^{2} \) | 1.67.g |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.75597759941283, −14.30595912220177, −13.43009539198130, −13.13993364003008, −12.92892073476343, −12.41645419989835, −11.73077309319137, −11.04489645307970, −10.53393879494452, −10.15864748218325, −9.480766067911308, −9.312265584030523, −8.474069011266284, −8.015557882457059, −7.544481481204108, −7.012065800875322, −6.363017685028261, −5.782089257705751, −5.333534664066693, −4.454847294400461, −4.031192659126795, −3.193480666601725, −2.840375085675020, −2.204375656109237, −1.485055169330133, 0, 0,
1.485055169330133, 2.204375656109237, 2.840375085675020, 3.193480666601725, 4.031192659126795, 4.454847294400461, 5.333534664066693, 5.782089257705751, 6.363017685028261, 7.012065800875322, 7.544481481204108, 8.015557882457059, 8.474069011266284, 9.312265584030523, 9.480766067911308, 10.15864748218325, 10.53393879494452, 11.04489645307970, 11.73077309319137, 12.41645419989835, 12.92892073476343, 13.13993364003008, 13.43009539198130, 14.30595912220177, 14.75597759941283
