| L(s) = 1 | + 3-s − 7-s + 9-s + 2·11-s + 13-s + 4·17-s + 8·19-s − 21-s + 27-s + 2·29-s − 10·31-s + 2·33-s − 2·37-s + 39-s + 12·41-s + 6·43-s − 4·47-s + 49-s + 4·51-s + 12·53-s + 8·57-s − 2·61-s − 63-s + 8·67-s − 2·73-s − 2·77-s − 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.970·17-s + 1.83·19-s − 0.218·21-s + 0.192·27-s + 0.371·29-s − 1.79·31-s + 0.348·33-s − 0.328·37-s + 0.160·39-s + 1.87·41-s + 0.914·43-s − 0.583·47-s + 1/7·49-s + 0.560·51-s + 1.64·53-s + 1.05·57-s − 0.256·61-s − 0.125·63-s + 0.977·67-s − 0.234·73-s − 0.227·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 218400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 218400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.734486792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.734486792\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−13.06104367390690, −12.49579884783689, −12.12366379745740, −11.59605383755286, −11.20322512503798, −10.53056347035441, −10.14553582448917, −9.556922189468718, −9.199238832830233, −8.963879532049244, −8.202809593303258, −7.685872727178018, −7.333050912431092, −6.949664580118180, −6.208957258097089, −5.669911813166746, −5.373753939373351, −4.636103857083057, −3.915021007124353, −3.602236149079326, −3.105194873606447, −2.521189262614281, −1.803466972945169, −1.108666146894651, −0.6626937722762362,
0.6626937722762362, 1.108666146894651, 1.803466972945169, 2.521189262614281, 3.105194873606447, 3.602236149079326, 3.915021007124353, 4.636103857083057, 5.373753939373351, 5.669911813166746, 6.208957258097089, 6.949664580118180, 7.333050912431092, 7.685872727178018, 8.202809593303258, 8.963879532049244, 9.199238832830233, 9.556922189468718, 10.14553582448917, 10.53056347035441, 11.20322512503798, 11.59605383755286, 12.12366379745740, 12.49579884783689, 13.06104367390690
