| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s + 13-s + 15-s − 2·17-s + 4·19-s − 4·21-s − 8·23-s + 25-s + 27-s − 2·29-s + 8·31-s − 4·35-s − 2·37-s + 39-s − 6·41-s + 12·43-s + 45-s + 9·49-s − 2·51-s − 10·53-s + 4·57-s + 10·61-s − 4·63-s + 65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.676·35-s − 0.328·37-s + 0.160·39-s − 0.937·41-s + 1.82·43-s + 0.149·45-s + 9/7·49-s − 0.280·51-s − 1.37·53-s + 0.529·57-s + 1.28·61-s − 0.503·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−16.23176922031963, −16.10181003369292, −15.61410409853885, −14.97602166545952, −14.04720115697401, −13.82700255340200, −13.35757993837136, −12.61425169998317, −12.28125171910284, −11.50568504070561, −10.69661491192813, −10.03600635705624, −9.631878531089523, −9.254993672469535, −8.410788387947862, −7.891637342434768, −7.025030449502729, −6.495621821995770, −5.969673268479365, −5.234151955742182, −4.191454584305073, −3.658292831884688, −2.876899965339973, −2.307090053138652, −1.226592771773968, 0,
1.226592771773968, 2.307090053138652, 2.876899965339973, 3.658292831884688, 4.191454584305073, 5.234151955742182, 5.969673268479365, 6.495621821995770, 7.025030449502729, 7.891637342434768, 8.410788387947862, 9.254993672469535, 9.631878531089523, 10.03600635705624, 10.69661491192813, 11.50568504070561, 12.28125171910284, 12.61425169998317, 13.35757993837136, 13.82700255340200, 14.04720115697401, 14.97602166545952, 15.61410409853885, 16.10181003369292, 16.23176922031963
