| L(s) = 1 | − 2·2-s − 3-s + 2·4-s + 2·6-s + 3·7-s + 9-s − 5·11-s − 2·12-s − 13-s − 6·14-s − 4·16-s − 5·17-s − 2·18-s + 2·19-s − 3·21-s + 10·22-s + 23-s + 2·26-s − 27-s + 6·28-s + 10·29-s − 2·31-s + 8·32-s + 5·33-s + 10·34-s + 2·36-s + 3·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s + 0.816·6-s + 1.13·7-s + 1/3·9-s − 1.50·11-s − 0.577·12-s − 0.277·13-s − 1.60·14-s − 16-s − 1.21·17-s − 0.471·18-s + 0.458·19-s − 0.654·21-s + 2.13·22-s + 0.208·23-s + 0.392·26-s − 0.192·27-s + 1.13·28-s + 1.85·29-s − 0.359·31-s + 1.41·32-s + 0.870·33-s + 1.71·34-s + 1/3·36-s + 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−9.701695706512646903485719857934, −8.601901886211211189403446744067, −8.084242246987417548623306093779, −7.36153153818067797309191660906, −6.45249708944222678026047767274, −5.06916684681045607590528795326, −4.63606321997536781672386123440, −2.62937015502897623552127991869, −1.48325443600957610417560300046, 0,
1.48325443600957610417560300046, 2.62937015502897623552127991869, 4.63606321997536781672386123440, 5.06916684681045607590528795326, 6.45249708944222678026047767274, 7.36153153818067797309191660906, 8.084242246987417548623306093779, 8.601901886211211189403446744067, 9.701695706512646903485719857934
