| L(s) = 1 | + 3-s − 4·7-s + 9-s + 13-s − 7·19-s − 4·21-s − 4·23-s − 5·25-s + 27-s + 6·29-s + 6·31-s − 3·37-s + 39-s − 5·41-s + 2·43-s + 7·47-s + 9·49-s − 2·53-s − 7·57-s − 14·59-s + 6·61-s − 4·63-s − 7·67-s − 4·69-s − 8·71-s − 16·73-s − 5·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.277·13-s − 1.60·19-s − 0.872·21-s − 0.834·23-s − 25-s + 0.192·27-s + 1.11·29-s + 1.07·31-s − 0.493·37-s + 0.160·39-s − 0.780·41-s + 0.304·43-s + 1.02·47-s + 9/7·49-s − 0.274·53-s − 0.927·57-s − 1.82·59-s + 0.768·61-s − 0.503·63-s − 0.855·67-s − 0.481·69-s − 0.949·71-s − 1.87·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−13.54492646636336, −12.92545558897002, −12.53805252503072, −12.01804954124345, −11.77281871710424, −10.79454838035004, −10.46290743026638, −10.05531071117446, −9.684948581567559, −9.024227540129261, −8.698559721796716, −8.235766830099139, −7.629819691479965, −7.140911746257078, −6.429765768717144, −6.227906205402227, −5.862948672568000, −4.893344240215533, −4.310932977799363, −3.955858587282667, −3.251542304097889, −2.909606008198896, −2.207311067106082, −1.702250585177144, −0.6871932117399980, 0,
0.6871932117399980, 1.702250585177144, 2.207311067106082, 2.909606008198896, 3.251542304097889, 3.955858587282667, 4.310932977799363, 4.893344240215533, 5.862948672568000, 6.227906205402227, 6.429765768717144, 7.140911746257078, 7.629819691479965, 8.235766830099139, 8.698559721796716, 9.024227540129261, 9.684948581567559, 10.05531071117446, 10.46290743026638, 10.79454838035004, 11.77281871710424, 12.01804954124345, 12.53805252503072, 12.92545558897002, 13.54492646636336
