| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 3·11-s + 13-s + 15-s − 2·17-s − 19-s + 21-s − 4·25-s + 27-s + 29-s + 6·31-s + 3·33-s + 35-s + 8·37-s + 39-s − 4·41-s + 45-s − 3·47-s − 6·49-s − 2·51-s + 3·55-s − 57-s + 4·59-s + 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s + 0.258·15-s − 0.485·17-s − 0.229·19-s + 0.218·21-s − 4/5·25-s + 0.192·27-s + 0.185·29-s + 1.07·31-s + 0.522·33-s + 0.169·35-s + 1.31·37-s + 0.160·39-s − 0.624·41-s + 0.149·45-s − 0.437·47-s − 6/7·49-s − 0.280·51-s + 0.404·55-s − 0.132·57-s + 0.520·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−12.98722992553666, −12.24392511460934, −11.79438191715291, −11.47265760298622, −11.02046709335030, −10.40248859940146, −9.876706633758867, −9.693807194660036, −9.055798650972515, −8.687885286994746, −8.223729943840045, −7.843842131458590, −7.249171171545766, −6.652074942687173, −6.352908599377058, −5.874910416779855, −5.244106901066442, −4.579337490461693, −4.323424877138171, −3.668003869876897, −3.234470866987330, −2.423890798448699, −2.162269174250493, −1.391877583451615, −1.020406515372356, 0,
1.020406515372356, 1.391877583451615, 2.162269174250493, 2.423890798448699, 3.234470866987330, 3.668003869876897, 4.323424877138171, 4.579337490461693, 5.244106901066442, 5.874910416779855, 6.352908599377058, 6.652074942687173, 7.249171171545766, 7.843842131458590, 8.223729943840045, 8.687885286994746, 9.055798650972515, 9.693807194660036, 9.876706633758867, 10.40248859940146, 11.02046709335030, 11.47265760298622, 11.79438191715291, 12.24392511460934, 12.98722992553666
