| L(s) = 1 | + 2-s − 3-s + 4-s + 4·5-s − 6-s − 7-s + 8-s + 9-s + 4·10-s − 12-s + 2·13-s − 14-s − 4·15-s + 16-s + 2·17-s + 18-s + 4·20-s + 21-s − 2·23-s − 24-s + 11·25-s + 2·26-s − 27-s − 28-s − 29-s − 4·30-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.288·12-s + 0.554·13-s − 0.267·14-s − 1.03·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.894·20-s + 0.218·21-s − 0.417·23-s − 0.204·24-s + 11/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s − 0.185·29-s − 0.730·30-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.918216407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.918216407\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−9.832359452926658863807637280539, −9.227113325636047270776722364526, −7.999930110461028592788411638480, −6.74954964488719477527454933713, −6.23655095662503498845373522731, −5.58652725497891668467484433545, −4.87114883253462968756112118545, −3.58224480636189118791782163461, −2.41793754232266905025946946351, −1.35941558913978044925336064577,
1.35941558913978044925336064577, 2.41793754232266905025946946351, 3.58224480636189118791782163461, 4.87114883253462968756112118545, 5.58652725497891668467484433545, 6.23655095662503498845373522731, 6.74954964488719477527454933713, 7.999930110461028592788411638480, 9.227113325636047270776722364526, 9.832359452926658863807637280539
