| L(s) = 1 | − 7-s − 13-s − 6·17-s − 5·19-s − 6·23-s + 6·29-s − 5·31-s + 7·37-s + 6·41-s − 43-s − 6·47-s − 6·49-s + 12·53-s − 61-s − 4·67-s + 6·71-s + 13·73-s − 5·79-s + 91-s + 13·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 6·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.277·13-s − 1.45·17-s − 1.14·19-s − 1.25·23-s + 1.11·29-s − 0.898·31-s + 1.15·37-s + 0.937·41-s − 0.152·43-s − 0.875·47-s − 6/7·49-s + 1.64·53-s − 0.128·61-s − 0.488·67-s + 0.712·71-s + 1.52·73-s − 0.562·79-s + 0.104·91-s + 1.31·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.550·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−13.54937516070244, −13.16348373433737, −12.68138520881230, −12.35394966702408, −11.66925701715733, −11.26282442360485, −10.77348857896164, −10.29919019328260, −9.772903467296144, −9.349522898635889, −8.730422328430113, −8.380549022280818, −7.821138770360544, −7.247796410193859, −6.592401000008550, −6.333858023653271, −5.838839624044781, −5.058223925215747, −4.520240291600638, −4.087977325719034, −3.556169075917480, −2.662797265615073, −2.297559389292481, −1.712563131203697, −0.6779800108230339, 0,
0.6779800108230339, 1.712563131203697, 2.297559389292481, 2.662797265615073, 3.556169075917480, 4.087977325719034, 4.520240291600638, 5.058223925215747, 5.838839624044781, 6.333858023653271, 6.592401000008550, 7.247796410193859, 7.821138770360544, 8.380549022280818, 8.730422328430113, 9.349522898635889, 9.772903467296144, 10.29919019328260, 10.77348857896164, 11.26282442360485, 11.66925701715733, 12.35394966702408, 12.68138520881230, 13.16348373433737, 13.54937516070244
