| L(s) = 1 | − 3-s + 5-s − 2·7-s − 2·9-s − 11-s − 15-s − 2·17-s + 2·21-s + 23-s − 4·25-s + 5·27-s + 7·31-s + 33-s − 2·35-s + 3·37-s + 8·41-s − 6·43-s − 2·45-s + 8·47-s − 3·49-s + 2·51-s + 6·53-s − 55-s − 5·59-s − 12·61-s + 4·63-s + 7·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s − 2/3·9-s − 0.301·11-s − 0.258·15-s − 0.485·17-s + 0.436·21-s + 0.208·23-s − 4/5·25-s + 0.962·27-s + 1.25·31-s + 0.174·33-s − 0.338·35-s + 0.493·37-s + 1.24·41-s − 0.914·43-s − 0.298·45-s + 1.16·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s − 0.134·55-s − 0.650·59-s − 1.53·61-s + 0.503·63-s + 0.855·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.144389907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.144389907\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−13.51984113854904, −13.15120446782469, −12.54375437835700, −12.17211500343377, −11.59417689893465, −11.19352451707968, −10.68566836666102, −10.12182674455324, −9.815987527814323, −9.119093268663625, −8.822178169572046, −8.132825190772583, −7.618859619409060, −7.014140217799353, −6.342111545636747, −6.070145964807760, −5.707587679874864, −4.918391762891497, −4.563304504125741, −3.769910403383720, −3.119460513127588, −2.587551951426347, −2.052843378609523, −1.056189188016063, −0.3785624053042284,
0.3785624053042284, 1.056189188016063, 2.052843378609523, 2.587551951426347, 3.119460513127588, 3.769910403383720, 4.563304504125741, 4.918391762891497, 5.707587679874864, 6.070145964807760, 6.342111545636747, 7.014140217799353, 7.618859619409060, 8.132825190772583, 8.822178169572046, 9.119093268663625, 9.815987527814323, 10.12182674455324, 10.68566836666102, 11.19352451707968, 11.59417689893465, 12.17211500343377, 12.54375437835700, 13.15120446782469, 13.51984113854904
