| L(s) = 1 | + 3-s − 3·5-s + 3·7-s − 2·9-s + 13-s − 3·15-s − 7·17-s + 4·19-s + 3·21-s + 4·23-s + 4·25-s − 5·27-s − 4·29-s + 8·31-s − 9·35-s − 7·37-s + 39-s − 2·41-s − 43-s + 6·45-s + 7·47-s + 2·49-s − 7·51-s − 4·53-s + 4·57-s − 14·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1.13·7-s − 2/3·9-s + 0.277·13-s − 0.774·15-s − 1.69·17-s + 0.917·19-s + 0.654·21-s + 0.834·23-s + 4/5·25-s − 0.962·27-s − 0.742·29-s + 1.43·31-s − 1.52·35-s − 1.15·37-s + 0.160·39-s − 0.312·41-s − 0.152·43-s + 0.894·45-s + 1.02·47-s + 2/7·49-s − 0.980·51-s − 0.549·53-s + 0.529·57-s − 1.82·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−8.310806280710314781206564093512, −7.64318637987977818131477606191, −7.06811394877393288371205798741, −5.99466371925698657398657931777, −4.91564381565398856402447843058, −4.40378187442761773937240109967, −3.47179110284320589858556541941, −2.70026531911479614580719171273, −1.51108618218951735364125857438, 0,
1.51108618218951735364125857438, 2.70026531911479614580719171273, 3.47179110284320589858556541941, 4.40378187442761773937240109967, 4.91564381565398856402447843058, 5.99466371925698657398657931777, 7.06811394877393288371205798741, 7.64318637987977818131477606191, 8.310806280710314781206564093512
