| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 4·7-s + 8-s + 9-s − 10-s − 4·11-s + 12-s + 7·13-s − 4·14-s − 15-s + 16-s + 5·17-s + 18-s − 20-s − 4·21-s − 4·22-s + 23-s + 24-s + 25-s + 7·26-s + 27-s − 4·28-s − 3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.288·12-s + 1.94·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 1.21·17-s + 0.235·18-s − 0.223·20-s − 0.872·21-s − 0.852·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s + 1.37·26-s + 0.192·27-s − 0.755·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−14.77622645016642, −14.19185027597937, −13.60680545524273, −13.15575620893311, −12.87111229221699, −12.55591305680825, −11.75000246086614, −11.11765211252233, −10.75621526191647, −10.11305407410229, −9.606204403789862, −9.100494689337678, −8.254097355622750, −8.023418676914339, −7.355712121366649, −6.660952347035080, −6.311622686124468, −5.499015278440111, −5.264109606011873, −4.140664404163507, −3.656783682581561, −3.271036712670759, −2.861048650136938, −1.908480505443379, −1.046203704796895, 0,
1.046203704796895, 1.908480505443379, 2.861048650136938, 3.271036712670759, 3.656783682581561, 4.140664404163507, 5.264109606011873, 5.499015278440111, 6.311622686124468, 6.660952347035080, 7.355712121366649, 8.023418676914339, 8.254097355622750, 9.100494689337678, 9.606204403789862, 10.11305407410229, 10.75621526191647, 11.11765211252233, 11.75000246086614, 12.55591305680825, 12.87111229221699, 13.15575620893311, 13.60680545524273, 14.19185027597937, 14.77622645016642
