| L(s) = 1 | − 7-s + 6·11-s − 4·13-s − 19-s + 6·23-s − 5·25-s + 3·29-s − 5·31-s − 2·37-s + 6·41-s + 8·43-s + 3·47-s + 49-s + 3·53-s + 15·59-s − 8·61-s − 4·67-s − 12·71-s − 8·73-s − 6·77-s − 2·79-s − 9·83-s + 4·91-s − 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.80·11-s − 1.10·13-s − 0.229·19-s + 1.25·23-s − 25-s + 0.557·29-s − 0.898·31-s − 0.328·37-s + 0.937·41-s + 1.21·43-s + 0.437·47-s + 1/7·49-s + 0.412·53-s + 1.95·59-s − 1.02·61-s − 0.488·67-s − 1.42·71-s − 0.936·73-s − 0.683·77-s − 0.225·79-s − 0.987·83-s + 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.45973030099069, −13.87940944317372, −13.38580939579976, −12.76700810897724, −12.26264167160649, −11.98241249822155, −11.36038030611663, −10.92403677928637, −10.25010687841859, −9.698153035732354, −9.283840406891865, −8.902151377160707, −8.350331958732167, −7.409972754516684, −7.188444574292629, −6.704394141697858, −5.950368028729981, −5.628009571265122, −4.780507940376070, −4.165299387466051, −3.849221353735767, −2.976999959644757, −2.461061066084816, −1.603490512261599, −0.9688327408575955, 0,
0.9688327408575955, 1.603490512261599, 2.461061066084816, 2.976999959644757, 3.849221353735767, 4.165299387466051, 4.780507940376070, 5.628009571265122, 5.950368028729981, 6.704394141697858, 7.188444574292629, 7.409972754516684, 8.350331958732167, 8.902151377160707, 9.283840406891865, 9.698153035732354, 10.25010687841859, 10.92403677928637, 11.36038030611663, 11.98241249822155, 12.26264167160649, 12.76700810897724, 13.38580939579976, 13.87940944317372, 14.45973030099069
