| L(s) = 1 | − 5-s + 4·7-s + 11-s + 2·13-s + 3·17-s + 19-s + 2·23-s − 4·25-s − 6·29-s − 8·31-s − 4·35-s + 6·37-s + 43-s − 2·47-s + 9·49-s + 2·53-s − 55-s + 2·59-s + 61-s − 2·65-s − 10·67-s − 71-s − 4·73-s + 4·77-s − 4·79-s − 5·83-s − 3·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s + 0.301·11-s + 0.554·13-s + 0.727·17-s + 0.229·19-s + 0.417·23-s − 4/5·25-s − 1.11·29-s − 1.43·31-s − 0.676·35-s + 0.986·37-s + 0.152·43-s − 0.291·47-s + 9/7·49-s + 0.274·53-s − 0.134·55-s + 0.260·59-s + 0.128·61-s − 0.248·65-s − 1.22·67-s − 0.118·71-s − 0.468·73-s + 0.455·77-s − 0.450·79-s − 0.548·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 229 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 5 T + p T^{2} \) | 1.83.f |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−13.71041277751587, −13.25628075244865, −12.77126006377140, −12.12956745790746, −11.69128542977765, −11.32408409185369, −10.97286850703548, −10.51067348280349, −9.751220898551613, −9.338188206358649, −8.754112326146199, −8.280982660289391, −7.814137478706874, −7.404675126240532, −7.017321512641613, −6.105678057794111, −5.593537917501016, −5.311771931588142, −4.485465098997554, −4.130384244285834, −3.570129141976518, −2.936290390258475, −2.046753440220587, −1.556208688799363, −1.007850889035494, 0,
1.007850889035494, 1.556208688799363, 2.046753440220587, 2.936290390258475, 3.570129141976518, 4.130384244285834, 4.485465098997554, 5.311771931588142, 5.593537917501016, 6.105678057794111, 7.017321512641613, 7.404675126240532, 7.814137478706874, 8.280982660289391, 8.754112326146199, 9.338188206358649, 9.751220898551613, 10.51067348280349, 10.97286850703548, 11.32408409185369, 11.69128542977765, 12.12956745790746, 12.77126006377140, 13.25628075244865, 13.71041277751587
