| L(s) = 1 | + 2-s − 4-s + 4·5-s − 3·8-s + 4·10-s + 2·11-s − 4·13-s − 16-s + 19-s − 4·20-s + 2·22-s + 6·23-s + 11·25-s − 4·26-s − 10·29-s + 5·32-s + 6·37-s + 38-s − 12·40-s − 10·41-s + 8·43-s − 2·44-s + 6·46-s + 12·47-s + 11·50-s + 4·52-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.78·5-s − 1.06·8-s + 1.26·10-s + 0.603·11-s − 1.10·13-s − 1/4·16-s + 0.229·19-s − 0.894·20-s + 0.426·22-s + 1.25·23-s + 11/5·25-s − 0.784·26-s − 1.85·29-s + 0.883·32-s + 0.986·37-s + 0.162·38-s − 1.89·40-s − 1.56·41-s + 1.21·43-s − 0.301·44-s + 0.884·46-s + 1.75·47-s + 1.55·50-s + 0.554·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.490923222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.490923222\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−7.62390698277960589116901759010, −6.91457115505992463731841411358, −6.18597915555350350519197372944, −5.57708007333785478308931483209, −5.14703563141717150666706163818, −4.44647749824266713401383734429, −3.50390508110879172883826336883, −2.66009560779142019458281938128, −1.96701635247693230351363417113, −0.835324140676339830468784137488,
0.835324140676339830468784137488, 1.96701635247693230351363417113, 2.66009560779142019458281938128, 3.50390508110879172883826336883, 4.44647749824266713401383734429, 5.14703563141717150666706163818, 5.57708007333785478308931483209, 6.18597915555350350519197372944, 6.91457115505992463731841411358, 7.62390698277960589116901759010
