| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s − 7-s + 9-s + 3·11-s − 2·12-s + 2·14-s − 4·16-s + 3·17-s − 2·18-s + 19-s + 21-s − 6·22-s + 23-s − 27-s − 2·28-s + 8·29-s − 10·31-s + 8·32-s − 3·33-s − 6·34-s + 2·36-s + 5·37-s − 2·38-s + 5·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.577·12-s + 0.534·14-s − 16-s + 0.727·17-s − 0.471·18-s + 0.229·19-s + 0.218·21-s − 1.27·22-s + 0.208·23-s − 0.192·27-s − 0.377·28-s + 1.48·29-s − 1.79·31-s + 1.41·32-s − 0.522·33-s − 1.02·34-s + 1/3·36-s + 0.821·37-s − 0.324·38-s + 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.007673089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007673089\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 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
−12.74881580976207, −12.43181183308633, −11.69267724700918, −11.31960975629162, −11.09783538982358, −10.39820367025600, −10.00028174294062, −9.690808907004034, −9.130361117491980, −8.869042471161891, −8.216574874935014, −7.745412931181386, −7.274411108454939, −6.832012400655256, −6.393571164908127, −5.830046917218531, −5.320520798855051, −4.641256336752163, −4.099715037425161, −3.582077274858711, −2.808534501860718, −2.208123105040234, −1.429904941454692, −0.9965444715769285, −0.4533039007190376,
0.4533039007190376, 0.9965444715769285, 1.429904941454692, 2.208123105040234, 2.808534501860718, 3.582077274858711, 4.099715037425161, 4.641256336752163, 5.320520798855051, 5.830046917218531, 6.393571164908127, 6.832012400655256, 7.274411108454939, 7.745412931181386, 8.216574874935014, 8.869042471161891, 9.130361117491980, 9.690808907004034, 10.00028174294062, 10.39820367025600, 11.09783538982358, 11.31960975629162, 11.69267724700918, 12.43181183308633, 12.74881580976207
