| L(s) = 1 | − 3·11-s + 13-s + 7·17-s − 19-s − 4·23-s − 4·29-s + 10·31-s + 12·37-s + 5·41-s − 12·43-s − 4·47-s − 7·49-s − 6·53-s − 4·59-s + 4·61-s + 5·67-s − 11·73-s − 4·79-s + 15·83-s + 11·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.904·11-s + 0.277·13-s + 1.69·17-s − 0.229·19-s − 0.834·23-s − 0.742·29-s + 1.79·31-s + 1.97·37-s + 0.780·41-s − 1.82·43-s − 0.583·47-s − 49-s − 0.824·53-s − 0.520·59-s + 0.512·61-s + 0.610·67-s − 1.28·73-s − 0.450·79-s + 1.64·83-s + 1.16·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 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
−14.68366374102381, −14.55164229573623, −13.76609992878994, −13.17979022402032, −13.02480625197147, −12.09755380222185, −11.94292244893337, −11.21771976686916, −10.70069628055506, −10.06406810228742, −9.758510528657066, −9.252275801893631, −8.227436285638482, −7.967853961022307, −7.759622694852187, −6.728266966712345, −6.281712100541399, −5.668110777228821, −5.145543854436634, −4.496204131322845, −3.834663273571341, −3.081082410373161, −2.651937705324853, −1.710677831347235, −0.9932909003355727, 0,
0.9932909003355727, 1.710677831347235, 2.651937705324853, 3.081082410373161, 3.834663273571341, 4.496204131322845, 5.145543854436634, 5.668110777228821, 6.281712100541399, 6.728266966712345, 7.759622694852187, 7.967853961022307, 8.227436285638482, 9.252275801893631, 9.758510528657066, 10.06406810228742, 10.70069628055506, 11.21771976686916, 11.94292244893337, 12.09755380222185, 13.02480625197147, 13.17979022402032, 13.76609992878994, 14.55164229573623, 14.68366374102381
