| L(s) = 1 | + 2·3-s − 7-s + 9-s + 6·19-s − 2·21-s − 23-s − 5·25-s − 4·27-s + 6·29-s − 8·31-s − 2·37-s + 2·41-s − 8·43-s − 8·47-s + 49-s − 2·53-s + 12·57-s − 6·59-s − 63-s − 12·67-s − 2·69-s + 8·71-s + 6·73-s − 10·75-s + 16·79-s − 11·81-s + 6·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.37·19-s − 0.436·21-s − 0.208·23-s − 25-s − 0.769·27-s + 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.312·41-s − 1.21·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s + 1.58·57-s − 0.781·59-s − 0.125·63-s − 1.46·67-s − 0.240·69-s + 0.949·71-s + 0.702·73-s − 1.15·75-s + 1.80·79-s − 1.22·81-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.76051156027558, −12.28870904811391, −11.86460049645981, −11.40946273332610, −10.94010145248957, −10.28510754960666, −9.887582106479610, −9.512883789816948, −9.041985365625468, −8.771085793482332, −8.015970045120654, −7.781123974401074, −7.468858176044408, −6.677436868474301, −6.393140969342122, −5.710168739191396, −5.216767285257696, −4.769625203567631, −3.996714534312330, −3.518593844198799, −3.221308819602526, −2.733864445038346, −1.959717455754100, −1.696078029067836, −0.7796547173830675, 0,
0.7796547173830675, 1.696078029067836, 1.959717455754100, 2.733864445038346, 3.221308819602526, 3.518593844198799, 3.996714534312330, 4.769625203567631, 5.216767285257696, 5.710168739191396, 6.393140969342122, 6.677436868474301, 7.468858176044408, 7.781123974401074, 8.015970045120654, 8.771085793482332, 9.041985365625468, 9.512883789816948, 9.887582106479610, 10.28510754960666, 10.94010145248957, 11.40946273332610, 11.86460049645981, 12.28870904811391, 12.76051156027558
