| L(s) = 1 | − 2-s + 4-s − 5-s − 3·7-s − 8-s + 10-s − 4·11-s − 5·13-s + 3·14-s + 16-s − 3·17-s − 5·19-s − 20-s + 4·22-s + 23-s + 25-s + 5·26-s − 3·28-s − 10·29-s + 6·31-s − 32-s + 3·34-s + 3·35-s − 2·37-s + 5·38-s + 40-s − 10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.13·7-s − 0.353·8-s + 0.316·10-s − 1.20·11-s − 1.38·13-s + 0.801·14-s + 1/4·16-s − 0.727·17-s − 1.14·19-s − 0.223·20-s + 0.852·22-s + 0.208·23-s + 1/5·25-s + 0.980·26-s − 0.566·28-s − 1.85·29-s + 1.07·31-s − 0.176·32-s + 0.514·34-s + 0.507·35-s − 0.328·37-s + 0.811·38-s + 0.158·40-s − 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 23 | \( 1 - T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−7.28282592620561521123593300726, −6.89915648435617772955562152789, −6.10835906950105789651198318344, −5.21020851711495748848795195399, −4.45867555484977030715795233337, −3.39303505922438412660104912747, −2.69112874409830102945728206957, −1.92228960395016464019532704735, 0, 0,
1.92228960395016464019532704735, 2.69112874409830102945728206957, 3.39303505922438412660104912747, 4.45867555484977030715795233337, 5.21020851711495748848795195399, 6.10835906950105789651198318344, 6.89915648435617772955562152789, 7.28282592620561521123593300726
