| L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 11-s + 4·13-s + 15-s − 4·19-s + 2·21-s + 4·23-s + 25-s − 27-s − 2·29-s − 8·31-s − 33-s + 2·35-s − 2·37-s − 4·39-s − 6·41-s − 6·43-s − 45-s − 4·47-s − 3·49-s + 6·53-s − 55-s + 4·57-s − 4·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 1.10·13-s + 0.258·15-s − 0.917·19-s + 0.436·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.174·33-s + 0.338·35-s − 0.328·37-s − 0.640·39-s − 0.937·41-s − 0.914·43-s − 0.149·45-s − 0.583·47-s − 3/7·49-s + 0.824·53-s − 0.134·55-s + 0.529·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−9.129193848306530308743004733501, −8.575695025320114472310807658399, −7.45032216727388844107431305007, −6.65440174622462091623221515291, −6.03712393732554439511406855181, −5.01885417498586988083749735645, −3.95226235353254144353386647622, −3.21897259288762576747115397853, −1.57464574379034617936342389770, 0,
1.57464574379034617936342389770, 3.21897259288762576747115397853, 3.95226235353254144353386647622, 5.01885417498586988083749735645, 6.03712393732554439511406855181, 6.65440174622462091623221515291, 7.45032216727388844107431305007, 8.575695025320114472310807658399, 9.129193848306530308743004733501
