| L(s) = 1 | − 5-s − 7-s + 4·13-s − 4·19-s + 25-s − 6·29-s + 10·31-s + 35-s + 8·37-s − 3·41-s − 43-s + 9·47-s − 6·49-s + 12·53-s + 6·59-s − 11·61-s − 4·65-s + 67-s − 6·71-s − 8·73-s + 14·79-s + 12·83-s + 15·89-s − 4·91-s + 4·95-s + 8·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.10·13-s − 0.917·19-s + 1/5·25-s − 1.11·29-s + 1.79·31-s + 0.169·35-s + 1.31·37-s − 0.468·41-s − 0.152·43-s + 1.31·47-s − 6/7·49-s + 1.64·53-s + 0.781·59-s − 1.40·61-s − 0.496·65-s + 0.122·67-s − 0.712·71-s − 0.936·73-s + 1.57·79-s + 1.31·83-s + 1.58·89-s − 0.419·91-s + 0.410·95-s + 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.090141875\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.090141875\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.68691517214110, −13.42073672815546, −13.02253430403023, −12.42087837946290, −11.85249034990213, −11.51155381972911, −10.91447404624763, −10.43211021249217, −10.05982438516584, −9.241561582988994, −8.931594242327001, −8.346675502600883, −7.886859999060216, −7.344770971532666, −6.692389525896355, −6.092858860801784, −5.948439056541514, −4.954255260708684, −4.483298909157678, −3.820619607951591, −3.463980978633605, −2.652219103479825, −2.079889207081630, −1.143333206595203, −0.5177603341895235,
0.5177603341895235, 1.143333206595203, 2.079889207081630, 2.652219103479825, 3.463980978633605, 3.820619607951591, 4.483298909157678, 4.954255260708684, 5.948439056541514, 6.092858860801784, 6.692389525896355, 7.344770971532666, 7.886859999060216, 8.346675502600883, 8.931594242327001, 9.241561582988994, 10.05982438516584, 10.43211021249217, 10.91447404624763, 11.51155381972911, 11.85249034990213, 12.42087837946290, 13.02253430403023, 13.42073672815546, 13.68691517214110
