| L(s) = 1 | − 2-s + 3-s − 4-s − 5-s − 6-s − 2·7-s + 3·8-s + 9-s + 10-s − 3·11-s − 12-s + 13-s + 2·14-s − 15-s − 16-s − 8·17-s − 18-s − 6·19-s + 20-s − 2·21-s + 3·22-s − 23-s + 3·24-s + 25-s − 26-s + 27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.755·7-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s − 0.258·15-s − 1/4·16-s − 1.94·17-s − 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.436·21-s + 0.639·22-s − 0.208·23-s + 0.612·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25215 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 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
−15.91306454591042, −15.37365236641446, −15.06991624392472, −14.15012630843534, −13.67384840885503, −13.26501373404325, −12.69655126269242, −12.44141968227344, −11.28447189689307, −10.84165445406571, −10.43746386759386, −9.754352685306236, −9.266951004435678, −8.600938958681014, −8.402868571838586, −7.804985685083700, −6.995527112446128, −6.635779560779005, −5.809614823367010, −4.841567762371926, −4.364726862158299, −3.849154225690965, −2.946136102872400, −2.296494878690535, −1.456086628785269, 0, 0,
1.456086628785269, 2.296494878690535, 2.946136102872400, 3.849154225690965, 4.364726862158299, 4.841567762371926, 5.809614823367010, 6.635779560779005, 6.995527112446128, 7.804985685083700, 8.402868571838586, 8.600938958681014, 9.266951004435678, 9.754352685306236, 10.43746386759386, 10.84165445406571, 11.28447189689307, 12.44141968227344, 12.69655126269242, 13.26501373404325, 13.67384840885503, 14.15012630843534, 15.06991624392472, 15.37365236641446, 15.91306454591042
