| L(s) = 1 | − 2·5-s − 4·7-s + 2·11-s + 6·13-s + 6·19-s − 25-s − 6·29-s − 4·31-s + 8·35-s + 2·37-s − 6·41-s + 2·43-s − 4·47-s + 9·49-s + 14·53-s − 4·55-s + 4·59-s − 10·61-s − 12·65-s − 2·67-s − 4·71-s + 2·73-s − 8·77-s + 12·79-s + 6·83-s − 12·89-s − 24·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 0.603·11-s + 1.66·13-s + 1.37·19-s − 1/5·25-s − 1.11·29-s − 0.718·31-s + 1.35·35-s + 0.328·37-s − 0.937·41-s + 0.304·43-s − 0.583·47-s + 9/7·49-s + 1.92·53-s − 0.539·55-s + 0.520·59-s − 1.28·61-s − 1.48·65-s − 0.244·67-s − 0.474·71-s + 0.234·73-s − 0.911·77-s + 1.35·79-s + 0.658·83-s − 1.27·89-s − 2.51·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−15.15837203558938, −14.77270742797632, −13.83785523839344, −13.47104539285586, −13.18646927722511, −12.39959748205317, −11.98282538853209, −11.48022698694455, −10.98357697935842, −10.39475063265644, −9.671585064189162, −9.270680712463015, −8.824434253820599, −8.084618851597953, −7.574158848303751, −6.874736385145476, −6.545421749165378, −5.742467956799258, −5.448903050615900, −4.266255078319839, −3.711467459241115, −3.504407001924212, −2.825982857487279, −1.678783603607788, −0.8846213970833806, 0,
0.8846213970833806, 1.678783603607788, 2.825982857487279, 3.504407001924212, 3.711467459241115, 4.266255078319839, 5.448903050615900, 5.742467956799258, 6.545421749165378, 6.874736385145476, 7.574158848303751, 8.084618851597953, 8.824434253820599, 9.270680712463015, 9.671585064189162, 10.39475063265644, 10.98357697935842, 11.48022698694455, 11.98282538853209, 12.39959748205317, 13.18646927722511, 13.47104539285586, 13.83785523839344, 14.77270742797632, 15.15837203558938
