| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s + 9-s + 10-s + 11-s − 12-s − 15-s + 16-s + 18-s + 5·19-s + 20-s + 22-s + 4·23-s − 24-s + 25-s − 27-s + 8·29-s − 30-s − 31-s + 32-s − 33-s + 36-s + 2·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.258·15-s + 1/4·16-s + 0.235·18-s + 1.14·19-s + 0.223·20-s + 0.213·22-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 1.48·29-s − 0.182·30-s − 0.179·31-s + 0.176·32-s − 0.174·33-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.472067409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.472067409\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−12.90568268052903, −12.22565523472952, −12.06859880339733, −11.56743913288588, −11.13953915817474, −10.57661670473755, −10.13361257192020, −9.825844408213694, −9.105086451684425, −8.763843796847681, −8.049154784920702, −7.573330758469077, −6.918979368545550, −6.642393042135581, −6.229988421423540, −5.489446379947679, −5.186969940762125, −4.853754231505463, −4.137745743859072, −3.592125185896440, −3.060755196236718, −2.476408284005009, −1.816217464699034, −1.106562087340531, −0.6560527363204549,
0.6560527363204549, 1.106562087340531, 1.816217464699034, 2.476408284005009, 3.060755196236718, 3.592125185896440, 4.137745743859072, 4.853754231505463, 5.186969940762125, 5.489446379947679, 6.229988421423540, 6.642393042135581, 6.918979368545550, 7.573330758469077, 8.049154784920702, 8.763843796847681, 9.105086451684425, 9.825844408213694, 10.13361257192020, 10.57661670473755, 11.13953915817474, 11.56743913288588, 12.06859880339733, 12.22565523472952, 12.90568268052903
