| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s + 4·13-s + 16-s − 17-s − 18-s − 3·19-s + 22-s − 23-s + 24-s − 5·25-s − 4·26-s − 27-s − 29-s + 6·31-s − 32-s + 33-s + 34-s + 36-s − 3·37-s + 3·38-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 − 0.301·11-s − 0.288·12-s + 1.10·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.688·19-s + 0.213·22-s − 0.208·23-s + 0.204·24-s − 25-s − 0.784·26-s − 0.192·27-s − 0.185·29-s + 1.07·31-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + 1/6·36-s − 0.493·37-s + 0.486·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−8.393204786253675776245515038412, −7.61586239923665589818112756679, −6.73464509674739036921063701998, −6.15613623133418553565160494715, −5.43415582309345200111188863667, −4.37986902344477161765997034638, −3.51748693863748698307944263465, −2.30718979324530668655650046506, −1.29710013516097103025668370108, 0,
1.29710013516097103025668370108, 2.30718979324530668655650046506, 3.51748693863748698307944263465, 4.37986902344477161765997034638, 5.43415582309345200111188863667, 6.15613623133418553565160494715, 6.73464509674739036921063701998, 7.61586239923665589818112756679, 8.393204786253675776245515038412
