| L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 2·10-s − 3·11-s − 4·16-s + 6·17-s + 6·19-s − 2·20-s − 6·22-s − 2·23-s + 25-s − 10·29-s − 4·31-s − 8·32-s + 12·34-s − 2·37-s + 12·38-s − 8·41-s − 2·43-s − 6·44-s − 4·46-s + 2·50-s + 14·53-s + 3·55-s − 20·58-s + 6·59-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 0.632·10-s − 0.904·11-s − 16-s + 1.45·17-s + 1.37·19-s − 0.447·20-s − 1.27·22-s − 0.417·23-s + 1/5·25-s − 1.85·29-s − 0.718·31-s − 1.41·32-s + 2.05·34-s − 0.328·37-s + 1.94·38-s − 1.24·41-s − 0.304·43-s − 0.904·44-s − 0.589·46-s + 0.282·50-s + 1.92·53-s + 0.404·55-s − 2.62·58-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−12.74755394419309, −12.33469347687660, −11.85714708542586, −11.61928580089440, −11.17383281971598, −10.49510944347248, −10.15655499825128, −9.604030798419448, −9.120332665321519, −8.588481627662215, −7.921526713146013, −7.595305874475092, −7.164396255306871, −6.733686231306513, −5.837178905441552, −5.636428406428758, −5.262412560812979, −4.881665047282537, −4.145450904313459, −3.625290225060366, −3.372002065257876, −2.909354959576328, −2.175165465024421, −1.642715847275204, −0.7414753973538094, 0,
0.7414753973538094, 1.642715847275204, 2.175165465024421, 2.909354959576328, 3.372002065257876, 3.625290225060366, 4.145450904313459, 4.881665047282537, 5.262412560812979, 5.636428406428758, 5.837178905441552, 6.733686231306513, 7.164396255306871, 7.595305874475092, 7.921526713146013, 8.588481627662215, 9.120332665321519, 9.604030798419448, 10.15655499825128, 10.49510944347248, 11.17383281971598, 11.61928580089440, 11.85714708542586, 12.33469347687660, 12.74755394419309
