| L(s) = 1 | − 3-s − 2·7-s + 9-s − 11-s + 4·13-s − 2·17-s + 2·21-s − 2·23-s − 5·25-s − 27-s − 2·29-s − 4·31-s + 33-s + 6·37-s − 4·39-s − 6·41-s − 12·43-s − 6·47-s − 3·49-s + 2·51-s − 2·63-s + 4·67-s + 2·69-s − 10·71-s + 2·73-s + 5·75-s + 2·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.485·17-s + 0.436·21-s − 0.417·23-s − 25-s − 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.174·33-s + 0.986·37-s − 0.640·39-s − 0.937·41-s − 1.82·43-s − 0.875·47-s − 3/7·49-s + 0.280·51-s − 0.251·63-s + 0.488·67-s + 0.240·69-s − 1.18·71-s + 0.234·73-s + 0.577·75-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−9.666664088585156836283870727400, −8.681228579284147789330769571895, −7.83184717123112089584609849816, −6.74842315612283598511124565736, −6.14928074186860196504069806728, −5.30108045271866710515332133457, −4.13249421192368957449183926090, −3.23862454287983130235067970596, −1.73357944322210682257057816907, 0,
1.73357944322210682257057816907, 3.23862454287983130235067970596, 4.13249421192368957449183926090, 5.30108045271866710515332133457, 6.14928074186860196504069806728, 6.74842315612283598511124565736, 7.83184717123112089584609849816, 8.681228579284147789330769571895, 9.666664088585156836283870727400
