| L(s) = 1 | + 3-s − 2·4-s + 3·7-s − 2·9-s − 5·11-s − 2·12-s − 4·13-s + 4·16-s + 4·17-s − 8·19-s + 3·21-s − 4·23-s − 5·27-s − 6·28-s + 4·29-s + 2·31-s − 5·33-s + 4·36-s − 37-s − 4·39-s − 5·41-s + 6·43-s + 10·44-s − 9·47-s + 4·48-s + 2·49-s + 4·51-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 1.13·7-s − 2/3·9-s − 1.50·11-s − 0.577·12-s − 1.10·13-s + 16-s + 0.970·17-s − 1.83·19-s + 0.654·21-s − 0.834·23-s − 0.962·27-s − 1.13·28-s + 0.742·29-s + 0.359·31-s − 0.870·33-s + 2/3·36-s − 0.164·37-s − 0.640·39-s − 0.780·41-s + 0.914·43-s + 1.50·44-s − 1.31·47-s + 0.577·48-s + 2/7·49-s + 0.560·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−9.691533618236397727139034060852, −8.471421016354941256803359955241, −8.222373016944275536672849141622, −7.57123135388662260649030454772, −5.94943989332909984100125223211, −5.04485018782635514558239582257, −4.47362183249546643029139662011, −3.10950868855398492563734235620, −2.06028069757458323954370127570, 0,
2.06028069757458323954370127570, 3.10950868855398492563734235620, 4.47362183249546643029139662011, 5.04485018782635514558239582257, 5.94943989332909984100125223211, 7.57123135388662260649030454772, 8.222373016944275536672849141622, 8.471421016354941256803359955241, 9.691533618236397727139034060852
