| L(s) = 1 | + 5-s − 2·7-s + 2·13-s + 6·17-s + 4·19-s + 6·23-s + 25-s − 6·29-s + 4·31-s − 2·35-s + 2·37-s − 6·41-s + 10·43-s − 6·47-s − 3·49-s + 6·53-s + 12·59-s + 2·61-s + 2·65-s − 2·67-s − 12·71-s + 2·73-s − 8·79-s + 6·83-s + 6·85-s + 6·89-s − 4·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.554·13-s + 1.45·17-s + 0.917·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.338·35-s + 0.328·37-s − 0.937·41-s + 1.52·43-s − 0.875·47-s − 3/7·49-s + 0.824·53-s + 1.56·59-s + 0.256·61-s + 0.248·65-s − 0.244·67-s − 1.42·71-s + 0.234·73-s − 0.900·79-s + 0.658·83-s + 0.650·85-s + 0.635·89-s − 0.419·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.630653748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.630653748\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 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.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−10.22754300302114228185381817663, −9.633261704500415779063377753491, −8.847850276232423443211930888771, −7.74818737522785607031159368119, −6.88656092970954199609923322632, −5.90138678053477774939394585997, −5.17606419420266207753384183426, −3.70610300459195515417889608627, −2.84705552122166969163585152299, −1.16915866227454488376692012454,
1.16915866227454488376692012454, 2.84705552122166969163585152299, 3.70610300459195515417889608627, 5.17606419420266207753384183426, 5.90138678053477774939394585997, 6.88656092970954199609923322632, 7.74818737522785607031159368119, 8.847850276232423443211930888771, 9.633261704500415779063377753491, 10.22754300302114228185381817663
