| L(s) = 1 | − 2·5-s + 6·11-s + 4·13-s − 6·17-s + 2·19-s − 25-s + 4·29-s + 8·31-s − 4·37-s − 6·41-s + 4·43-s − 7·49-s + 10·53-s − 12·55-s + 14·59-s + 12·61-s − 8·65-s + 2·67-s + 8·71-s + 10·73-s − 16·79-s − 14·83-s + 12·85-s + 6·89-s − 4·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.80·11-s + 1.10·13-s − 1.45·17-s + 0.458·19-s − 1/5·25-s + 0.742·29-s + 1.43·31-s − 0.657·37-s − 0.937·41-s + 0.609·43-s − 49-s + 1.37·53-s − 1.61·55-s + 1.82·59-s + 1.53·61-s − 0.992·65-s + 0.244·67-s + 0.949·71-s + 1.17·73-s − 1.80·79-s − 1.53·83-s + 1.30·85-s + 0.635·89-s − 0.410·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.808458008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.808458008\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 1129 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.32871255921759, −12.75886683493845, −12.17374125613018, −11.64176629689670, −11.53763583010586, −11.11171333684553, −10.46207875213478, −9.813440221625873, −9.500248860963860, −8.678685134692815, −8.431945244859177, −8.304852555703580, −7.229377491533754, −6.882688577997475, −6.568554926180570, −6.005929147744537, −5.362135880727613, −4.572043673725644, −4.195738044161537, −3.758290970378728, −3.336779541026578, −2.495151449622413, −1.802530851548842, −1.071363033579581, −0.5719679212339373,
0.5719679212339373, 1.071363033579581, 1.802530851548842, 2.495151449622413, 3.336779541026578, 3.758290970378728, 4.195738044161537, 4.572043673725644, 5.362135880727613, 6.005929147744537, 6.568554926180570, 6.882688577997475, 7.229377491533754, 8.304852555703580, 8.431945244859177, 8.678685134692815, 9.500248860963860, 9.813440221625873, 10.46207875213478, 11.11171333684553, 11.53763583010586, 11.64176629689670, 12.17374125613018, 12.75886683493845, 13.32871255921759
