| L(s) = 1 | + 5-s − 7-s − 11-s − 13-s − 4·17-s − 19-s + 2·23-s + 25-s − 2·29-s + 5·31-s − 35-s + 2·37-s − 12·41-s + 9·43-s + 9·47-s − 6·49-s + 8·53-s − 55-s + 9·59-s − 13·61-s − 65-s + 7·67-s + 14·71-s + 6·73-s + 77-s − 7·79-s + 5·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 0.301·11-s − 0.277·13-s − 0.970·17-s − 0.229·19-s + 0.417·23-s + 1/5·25-s − 0.371·29-s + 0.898·31-s − 0.169·35-s + 0.328·37-s − 1.87·41-s + 1.37·43-s + 1.31·47-s − 6/7·49-s + 1.09·53-s − 0.134·55-s + 1.17·59-s − 1.66·61-s − 0.124·65-s + 0.855·67-s + 1.66·71-s + 0.702·73-s + 0.113·77-s − 0.787·79-s + 0.548·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.870591650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.870591650\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−13.67645662857521, −13.21385424124517, −12.89852151062971, −12.21644306778115, −11.92366418666749, −11.07230252522519, −10.90285806548605, −10.20796055196142, −9.833503927956654, −9.280213685547967, −8.808418084255017, −8.332876200328460, −7.710877032139011, −7.097370619820916, −6.641899842299758, −6.208179984049301, −5.532288781006178, −5.057850433092903, −4.469825392419319, −3.867412508030433, −3.209998188359065, −2.445992888523482, −2.192432705653036, −1.224189823930599, −0.4409478680755035,
0.4409478680755035, 1.224189823930599, 2.192432705653036, 2.445992888523482, 3.209998188359065, 3.867412508030433, 4.469825392419319, 5.057850433092903, 5.532288781006178, 6.208179984049301, 6.641899842299758, 7.097370619820916, 7.710877032139011, 8.332876200328460, 8.808418084255017, 9.280213685547967, 9.833503927956654, 10.20796055196142, 10.90285806548605, 11.07230252522519, 11.92366418666749, 12.21644306778115, 12.89852151062971, 13.21385424124517, 13.67645662857521
