| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 4·11-s − 6·13-s + 16-s + 6·17-s − 8·19-s + 20-s − 4·22-s + 4·23-s + 25-s − 6·26-s − 10·29-s + 32-s + 6·34-s − 6·37-s − 8·38-s + 40-s + 6·41-s + 43-s − 4·44-s + 4·46-s − 12·47-s − 7·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 1.20·11-s − 1.66·13-s + 1/4·16-s + 1.45·17-s − 1.83·19-s + 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s − 1.17·26-s − 1.85·29-s + 0.176·32-s + 1.02·34-s − 0.986·37-s − 1.29·38-s + 0.158·40-s + 0.937·41-s + 0.152·43-s − 0.603·44-s + 0.589·46-s − 1.75·47-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−7.83236519144298405639922019001, −7.41703319083255995441325989809, −6.57098662113579271915044423728, −5.63063466756680016537932050708, −5.18274351014570875712329058767, −4.46235806637932515739992684905, −3.34654532412407599099351267724, −2.55926155390428591489412574957, −1.79556781925727990335298059867, 0,
1.79556781925727990335298059867, 2.55926155390428591489412574957, 3.34654532412407599099351267724, 4.46235806637932515739992684905, 5.18274351014570875712329058767, 5.63063466756680016537932050708, 6.57098662113579271915044423728, 7.41703319083255995441325989809, 7.83236519144298405639922019001
