| L(s) = 1 | − 3-s − 7-s + 9-s − 5·11-s + 4·13-s − 17-s + 19-s + 21-s − 27-s + 9·29-s − 6·31-s + 5·33-s − 3·37-s − 4·39-s + 5·41-s + 2·43-s + 9·47-s − 6·49-s + 51-s + 3·53-s − 57-s − 6·59-s − 63-s − 14·67-s − 8·71-s + 7·73-s + 5·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 1.10·13-s − 0.242·17-s + 0.229·19-s + 0.218·21-s − 0.192·27-s + 1.67·29-s − 1.07·31-s + 0.870·33-s − 0.493·37-s − 0.640·39-s + 0.780·41-s + 0.304·43-s + 1.31·47-s − 6/7·49-s + 0.140·51-s + 0.412·53-s − 0.132·57-s − 0.781·59-s − 0.125·63-s − 1.71·67-s − 0.949·71-s + 0.819·73-s + 0.569·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−7.80101498110629500414153987633, −7.15588662798600336752522189959, −6.29109778398469615881288493724, −5.74828815182154504555401334420, −5.02477989924825182865279003143, −4.22341223003544440578010036634, −3.26178520883866927668882298201, −2.45451092876252325005184007965, −1.20375695326824358810449285152, 0,
1.20375695326824358810449285152, 2.45451092876252325005184007965, 3.26178520883866927668882298201, 4.22341223003544440578010036634, 5.02477989924825182865279003143, 5.74828815182154504555401334420, 6.29109778398469615881288493724, 7.15588662798600336752522189959, 7.80101498110629500414153987633
