| L(s) = 1 | + 5-s + 7-s + 4·11-s + 13-s − 17-s − 2·19-s − 2·23-s − 4·25-s + 35-s − 5·37-s + 6·41-s − 9·43-s − 5·47-s − 6·49-s + 6·53-s + 4·55-s − 4·59-s − 10·61-s + 65-s + 2·67-s − 15·71-s − 4·73-s + 4·77-s − 4·79-s − 85-s − 10·89-s + 91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 1.20·11-s + 0.277·13-s − 0.242·17-s − 0.458·19-s − 0.417·23-s − 4/5·25-s + 0.169·35-s − 0.821·37-s + 0.937·41-s − 1.37·43-s − 0.729·47-s − 6/7·49-s + 0.824·53-s + 0.539·55-s − 0.520·59-s − 1.28·61-s + 0.124·65-s + 0.244·67-s − 1.78·71-s − 0.468·73-s + 0.455·77-s − 0.450·79-s − 0.108·85-s − 1.05·89-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−16.41970705366729, −15.76012207708361, −15.18534495418682, −14.53003961546362, −14.24038223784875, −13.50424774852607, −13.16327616534365, −12.30150058765376, −11.82957401294524, −11.33636576110763, −10.68459168536867, −10.03367960618256, −9.502318422390590, −8.858550794571048, −8.390858600067210, −7.651116428913282, −6.932432827849323, −6.291074211350972, −5.868384126708223, −5.003240873733802, −4.298905609325182, −3.723689069332969, −2.851234605663175, −1.817697582139484, −1.409909879608182, 0,
1.409909879608182, 1.817697582139484, 2.851234605663175, 3.723689069332969, 4.298905609325182, 5.003240873733802, 5.868384126708223, 6.291074211350972, 6.932432827849323, 7.651116428913282, 8.390858600067210, 8.858550794571048, 9.502318422390590, 10.03367960618256, 10.68459168536867, 11.33636576110763, 11.82957401294524, 12.30150058765376, 13.16327616534365, 13.50424774852607, 14.24038223784875, 14.53003961546362, 15.18534495418682, 15.76012207708361, 16.41970705366729
