| L(s) = 1 | − 2·5-s − 7-s + 2·11-s + 4·17-s − 4·19-s + 6·23-s − 25-s − 6·29-s + 31-s + 2·35-s + 10·37-s + 4·41-s − 43-s − 10·47-s − 6·49-s − 8·53-s − 4·55-s − 2·59-s − 5·61-s + 7·67-s + 10·71-s − 7·73-s − 2·77-s − 17·79-s + 12·83-s − 8·85-s + 16·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.377·7-s + 0.603·11-s + 0.970·17-s − 0.917·19-s + 1.25·23-s − 1/5·25-s − 1.11·29-s + 0.179·31-s + 0.338·35-s + 1.64·37-s + 0.624·41-s − 0.152·43-s − 1.45·47-s − 6/7·49-s − 1.09·53-s − 0.539·55-s − 0.260·59-s − 0.640·61-s + 0.855·67-s + 1.18·71-s − 0.819·73-s − 0.227·77-s − 1.91·79-s + 1.31·83-s − 0.867·85-s + 1.69·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−15.69296121866367, −15.06132302356423, −14.59302541374993, −14.37092521855953, −13.27241148829631, −13.00736887934775, −12.48598554855110, −11.81333730731395, −11.33997786497918, −10.96342901830401, −10.20149808805650, −9.482374741625786, −9.224104357757282, −8.378181352696633, −7.804909564758208, −7.475897724180447, −6.540814533703350, −6.276995225095236, −5.390210091558402, −4.680351707479287, −4.064152583510993, −3.444364066389142, −2.903431328633510, −1.859456902363824, −0.9691496682059339, 0,
0.9691496682059339, 1.859456902363824, 2.903431328633510, 3.444364066389142, 4.064152583510993, 4.680351707479287, 5.390210091558402, 6.276995225095236, 6.540814533703350, 7.475897724180447, 7.804909564758208, 8.378181352696633, 9.224104357757282, 9.482374741625786, 10.20149808805650, 10.96342901830401, 11.33997786497918, 11.81333730731395, 12.48598554855110, 13.00736887934775, 13.27241148829631, 14.37092521855953, 14.59302541374993, 15.06132302356423, 15.69296121866367
