| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 2·11-s + 12-s − 7·13-s + 16-s + 7·17-s + 18-s − 8·19-s − 2·22-s − 5·23-s + 24-s − 7·26-s + 27-s + 9·29-s − 31-s + 32-s − 2·33-s + 7·34-s + 36-s − 2·37-s − 8·38-s − 7·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 1.94·13-s + 1/4·16-s + 1.69·17-s + 0.235·18-s − 1.83·19-s − 0.426·22-s − 1.04·23-s + 0.204·24-s − 1.37·26-s + 0.192·27-s + 1.67·29-s − 0.179·31-s + 0.176·32-s − 0.348·33-s + 1.20·34-s + 1/6·36-s − 0.328·37-s − 1.29·38-s − 1.12·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.61578525849557810082041381311, −6.86459775322558522461574614804, −6.14968799483881721249827500251, −5.26075390625256239733437532333, −4.72131762994560594839583670413, −3.99684885775185847664073478878, −3.02655868394079864844594725672, −2.50203653191820691868489281428, −1.64511139213912030365153843556, 0,
1.64511139213912030365153843556, 2.50203653191820691868489281428, 3.02655868394079864844594725672, 3.99684885775185847664073478878, 4.72131762994560594839583670413, 5.26075390625256239733437532333, 6.14968799483881721249827500251, 6.86459775322558522461574614804, 7.61578525849557810082041381311
