| L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s + 4·7-s + 8-s + 9-s + 3·10-s − 3·11-s − 12-s + 2·13-s + 4·14-s − 3·15-s + 16-s + 18-s + 8·19-s + 3·20-s − 4·21-s − 3·22-s − 6·23-s − 24-s + 4·25-s + 2·26-s − 27-s + 4·28-s − 3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.904·11-s − 0.288·12-s + 0.554·13-s + 1.06·14-s − 0.774·15-s + 1/4·16-s + 0.235·18-s + 1.83·19-s + 0.670·20-s − 0.872·21-s − 0.639·22-s − 1.25·23-s − 0.204·24-s + 4/5·25-s + 0.392·26-s − 0.192·27-s + 0.755·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.393469534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.393469534\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−9.575141077697547186298976783305, −8.313421474211774661211358229128, −7.72289951773323931710937718925, −6.67241173567538214923332783179, −5.79967338035782189586611830648, −5.23887560284786213143454437893, −4.77207058101888659734681390414, −3.40304186691744279587696353930, −2.11306556814007985634557939865, −1.39533560657326728163111967124,
1.39533560657326728163111967124, 2.11306556814007985634557939865, 3.40304186691744279587696353930, 4.77207058101888659734681390414, 5.23887560284786213143454437893, 5.79967338035782189586611830648, 6.67241173567538214923332783179, 7.72289951773323931710937718925, 8.313421474211774661211358229128, 9.575141077697547186298976783305
