| L(s) = 1 | + 3·7-s − 3·9-s + 3·11-s + 4·17-s − 7·19-s + 4·23-s + 8·29-s − 10·31-s + 3·37-s − 2·41-s + 6·43-s + 47-s + 2·49-s − 9·53-s + 4·59-s + 14·61-s − 9·63-s + 4·67-s + 6·71-s − 4·73-s + 9·77-s + 10·79-s + 9·81-s + 89-s − 16·97-s − 9·99-s + 101-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 9-s + 0.904·11-s + 0.970·17-s − 1.60·19-s + 0.834·23-s + 1.48·29-s − 1.79·31-s + 0.493·37-s − 0.312·41-s + 0.914·43-s + 0.145·47-s + 2/7·49-s − 1.23·53-s + 0.520·59-s + 1.79·61-s − 1.13·63-s + 0.488·67-s + 0.712·71-s − 0.468·73-s + 1.02·77-s + 1.12·79-s + 81-s + 0.105·89-s − 1.62·97-s − 0.904·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−12.85987661451197, −12.51893943250784, −12.10719008274149, −11.47074997766669, −11.25296889434284, −10.81195520059663, −10.41717793034796, −9.706800261734376, −9.204273980656413, −8.802119727995659, −8.308807090010223, −8.048017490482245, −7.476533313704421, −6.758440601372121, −6.489878364160932, −5.853529108327988, −5.294481357756986, −5.001497814717708, −4.298813572052686, −3.854168752150533, −3.300827677841953, −2.551998486790861, −2.147443644165216, −1.379240546084428, −0.9169053771850081, 0,
0.9169053771850081, 1.379240546084428, 2.147443644165216, 2.551998486790861, 3.300827677841953, 3.854168752150533, 4.298813572052686, 5.001497814717708, 5.294481357756986, 5.853529108327988, 6.489878364160932, 6.758440601372121, 7.476533313704421, 8.048017490482245, 8.308807090010223, 8.802119727995659, 9.204273980656413, 9.706800261734376, 10.41717793034796, 10.81195520059663, 11.25296889434284, 11.47074997766669, 12.10719008274149, 12.51893943250784, 12.85987661451197
