| L(s) = 1 | − 3-s + 9-s + 4·11-s + 2·13-s + 2·17-s + 4·19-s − 8·23-s − 27-s − 6·29-s + 8·31-s − 4·33-s + 6·37-s − 2·39-s + 6·41-s − 4·43-s − 2·51-s − 2·53-s − 4·57-s − 4·59-s − 2·61-s + 4·67-s + 8·69-s − 8·71-s + 10·73-s + 8·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.280·51-s − 0.274·53-s − 0.529·57-s − 0.520·59-s − 0.256·61-s + 0.488·67-s + 0.963·69-s − 0.949·71-s + 1.17·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.31519200759433, −12.35132153378230, −12.25277806410578, −11.78057298073643, −11.30859776565314, −10.99084712419308, −10.32378979813217, −9.834094072477818, −9.466030811842965, −9.133506548513069, −8.291956093149011, −7.983296671535843, −7.504313891027125, −6.866059891273209, −6.337409624022965, −6.040868897262323, −5.569120083772533, −4.957091346556686, −4.288246436839316, −3.935396087012182, −3.439744845890568, −2.725617159337591, −1.983024710665593, −1.312483553352705, −0.9193964495701101, 0,
0.9193964495701101, 1.312483553352705, 1.983024710665593, 2.725617159337591, 3.439744845890568, 3.935396087012182, 4.288246436839316, 4.957091346556686, 5.569120083772533, 6.040868897262323, 6.337409624022965, 6.866059891273209, 7.504313891027125, 7.983296671535843, 8.291956093149011, 9.133506548513069, 9.466030811842965, 9.834094072477818, 10.32378979813217, 10.99084712419308, 11.30859776565314, 11.78057298073643, 12.25277806410578, 12.35132153378230, 13.31519200759433
