| L(s) = 1 | − 2-s − 3-s − 4-s + 6-s − 4·7-s + 3·8-s + 9-s − 4·11-s + 12-s − 13-s + 4·14-s − 16-s − 18-s + 4·21-s + 4·22-s − 3·24-s + 26-s − 27-s + 4·28-s + 10·29-s − 4·31-s − 5·32-s + 4·33-s − 36-s − 2·37-s + 39-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.408·6-s − 1.51·7-s + 1.06·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s − 0.277·13-s + 1.06·14-s − 1/4·16-s − 0.235·18-s + 0.872·21-s + 0.852·22-s − 0.612·24-s + 0.196·26-s − 0.192·27-s + 0.755·28-s + 1.85·29-s − 0.718·31-s − 0.883·32-s + 0.696·33-s − 1/6·36-s − 0.328·37-s + 0.160·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 281775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.88247461701405, −12.53497735958554, −12.28748382568769, −11.43044763882888, −11.06711177663676, −10.31556503668217, −10.20775122812756, −9.966565317729758, −9.199860080008009, −9.042230289131460, −8.269245240016640, −7.989354724573297, −7.347560356060492, −6.853078761507541, −6.579338562922888, −5.738919594206982, −5.502860197614630, −4.877671114045576, −4.400249397772672, −3.797415232006423, −3.202773206364286, −2.642564203276219, −2.055132568800481, −1.103469573287093, −0.5408719604895804, 0,
0.5408719604895804, 1.103469573287093, 2.055132568800481, 2.642564203276219, 3.202773206364286, 3.797415232006423, 4.400249397772672, 4.877671114045576, 5.502860197614630, 5.738919594206982, 6.579338562922888, 6.853078761507541, 7.347560356060492, 7.989354724573297, 8.269245240016640, 9.042230289131460, 9.199860080008009, 9.966565317729758, 10.20775122812756, 10.31556503668217, 11.06711177663676, 11.43044763882888, 12.28748382568769, 12.53497735958554, 12.88247461701405
