| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 4·7-s − 8-s + 9-s + 3·11-s − 12-s − 4·13-s − 4·14-s + 16-s − 3·17-s − 18-s + 7·19-s − 4·21-s − 3·22-s − 23-s + 24-s − 5·25-s + 4·26-s − 27-s + 4·28-s + 8·31-s − 32-s − 3·33-s + 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s − 1.10·13-s − 1.06·14-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 1.60·19-s − 0.872·21-s − 0.639·22-s − 0.208·23-s + 0.204·24-s − 25-s + 0.784·26-s − 0.192·27-s + 0.755·28-s + 1.43·31-s − 0.176·32-s − 0.522·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 255162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 255162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.041314299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.041314299\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 23 | \( 1 + T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−12.50734060436918, −12.13074339441855, −11.77539287522115, −11.43770741106707, −11.21185178541712, −10.35097263675138, −10.17255047911480, −9.601538531458971, −9.156930922196327, −8.625021845528165, −8.124483921734987, −7.667569076540573, −7.236680597879455, −6.858679526441222, −6.213029532400765, −5.639836503494043, −5.212556457836624, −4.583397011468668, −4.359621987969237, −3.523777996465990, −2.868626390284151, −2.029761658203546, −1.790377495410768, −1.038217444911894, −0.5174842970014336,
0.5174842970014336, 1.038217444911894, 1.790377495410768, 2.029761658203546, 2.868626390284151, 3.523777996465990, 4.359621987969237, 4.583397011468668, 5.212556457836624, 5.639836503494043, 6.213029532400765, 6.858679526441222, 7.236680597879455, 7.667569076540573, 8.124483921734987, 8.625021845528165, 9.156930922196327, 9.601538531458971, 10.17255047911480, 10.35097263675138, 11.21185178541712, 11.43770741106707, 11.77539287522115, 12.13074339441855, 12.50734060436918
