| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s − 11-s + 12-s − 16-s − 17-s + 18-s − 6·19-s − 22-s + 3·24-s − 5·25-s − 27-s − 10·29-s − 8·31-s + 5·32-s + 33-s − 34-s − 36-s + 6·37-s − 6·38-s − 6·41-s − 2·43-s + 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.37·19-s − 0.213·22-s + 0.612·24-s − 25-s − 0.192·27-s − 1.85·29-s − 1.43·31-s + 0.883·32-s + 0.174·33-s − 0.171·34-s − 1/6·36-s + 0.986·37-s − 0.973·38-s − 0.937·41-s − 0.304·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94809 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94809 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.96289925995304, −13.31199164128393, −13.14612499468281, −12.72944999280859, −12.25581284026737, −11.54892383281001, −11.25273496819168, −10.79367981481938, −9.961325671249885, −9.764730754417854, −9.074001799951922, −8.617502744585467, −8.020401591557703, −7.482566095728980, −6.791039174199506, −6.251346501510304, −5.818810140186904, −5.223464080879563, −4.907696803819331, −4.104462392869146, −3.794974352109438, −3.247940700077145, −2.145436789301594, −1.908859933603816, −0.6114117401104049, 0,
0.6114117401104049, 1.908859933603816, 2.145436789301594, 3.247940700077145, 3.794974352109438, 4.104462392869146, 4.907696803819331, 5.223464080879563, 5.818810140186904, 6.251346501510304, 6.791039174199506, 7.482566095728980, 8.020401591557703, 8.617502744585467, 9.074001799951922, 9.764730754417854, 9.961325671249885, 10.79367981481938, 11.25273496819168, 11.54892383281001, 12.25581284026737, 12.72944999280859, 13.14612499468281, 13.31199164128393, 13.96289925995304
