| L(s) = 1 | − 4·7-s + 11-s − 4·13-s − 2·17-s − 2·19-s − 2·23-s − 5·25-s + 10·29-s + 8·31-s − 37-s + 6·41-s + 6·43-s − 4·47-s + 9·49-s − 6·53-s + 10·59-s + 12·67-s − 8·71-s + 10·73-s − 4·77-s − 2·79-s + 12·83-s − 12·89-s + 16·91-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.301·11-s − 1.10·13-s − 0.485·17-s − 0.458·19-s − 0.417·23-s − 25-s + 1.85·29-s + 1.43·31-s − 0.164·37-s + 0.937·41-s + 0.914·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s + 1.30·59-s + 1.46·67-s − 0.949·71-s + 1.17·73-s − 0.455·77-s − 0.225·79-s + 1.31·83-s − 1.27·89-s + 1.67·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 234432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 234432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.480405732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.480405732\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 37 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−13.01127400674686, −12.36746726440377, −12.11165991539146, −11.66734660280686, −11.04792026379873, −10.44202171677114, −9.953141360793174, −9.788722889028592, −9.316786016255081, −8.707920948221728, −8.234588864378056, −7.717476364140860, −7.117816897421840, −6.605930436606107, −6.324071388039762, −5.909133076353155, −5.139608679706716, −4.587320071719526, −4.141972328815826, −3.569009672086189, −2.911308731519532, −2.496303590542869, −2.006033554633727, −0.9151392733442861, −0.4022196572235084,
0.4022196572235084, 0.9151392733442861, 2.006033554633727, 2.496303590542869, 2.911308731519532, 3.569009672086189, 4.141972328815826, 4.587320071719526, 5.139608679706716, 5.909133076353155, 6.324071388039762, 6.605930436606107, 7.117816897421840, 7.717476364140860, 8.234588864378056, 8.707920948221728, 9.316786016255081, 9.788722889028592, 9.953141360793174, 10.44202171677114, 11.04792026379873, 11.66734660280686, 12.11165991539146, 12.36746726440377, 13.01127400674686
