| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s + 11-s + 13-s − 4·14-s + 16-s − 4·19-s + 22-s − 5·25-s + 26-s − 4·28-s − 10·31-s + 32-s + 2·37-s − 4·38-s + 6·41-s − 10·43-s + 44-s + 9·49-s − 5·50-s + 52-s − 6·53-s − 4·56-s + 2·61-s − 10·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s + 0.301·11-s + 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.917·19-s + 0.213·22-s − 25-s + 0.196·26-s − 0.755·28-s − 1.79·31-s + 0.176·32-s + 0.328·37-s − 0.648·38-s + 0.937·41-s − 1.52·43-s + 0.150·44-s + 9/7·49-s − 0.707·50-s + 0.138·52-s − 0.824·53-s − 0.534·56-s + 0.256·61-s − 1.27·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−8.557035733463942560599715120987, −7.52104320390851123045764221154, −6.78175364777231647019603915327, −6.14149234771220156853905294128, −5.55677167990463394119827611669, −4.33367672755955602988575320655, −3.67086638494344903515072731439, −2.90553344815263021979503844085, −1.77465525514607936382065887676, 0,
1.77465525514607936382065887676, 2.90553344815263021979503844085, 3.67086638494344903515072731439, 4.33367672755955602988575320655, 5.55677167990463394119827611669, 6.14149234771220156853905294128, 6.78175364777231647019603915327, 7.52104320390851123045764221154, 8.557035733463942560599715120987
