| L(s) = 1 | − 3-s + 9-s − 2·11-s + 5·13-s − 2·17-s − 3·19-s − 2·23-s − 5·25-s − 27-s + 8·29-s + 31-s + 2·33-s − 5·37-s − 5·39-s + 2·41-s − 7·43-s + 8·47-s + 2·51-s − 2·53-s + 3·57-s − 10·59-s − 2·61-s + 11·67-s + 2·69-s − 12·71-s − 3·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s + 1.38·13-s − 0.485·17-s − 0.688·19-s − 0.417·23-s − 25-s − 0.192·27-s + 1.48·29-s + 0.179·31-s + 0.348·33-s − 0.821·37-s − 0.800·39-s + 0.312·41-s − 1.06·43-s + 1.16·47-s + 0.280·51-s − 0.274·53-s + 0.397·57-s − 1.30·59-s − 0.256·61-s + 1.34·67-s + 0.240·69-s − 1.42·71-s − 0.351·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.041459520984952611250563019274, −7.12849542224079283196468796051, −6.29211351940374144460927791725, −5.94332397504937360956820682604, −4.97069578950949997794214798605, −4.25317810266192389378879969849, −3.43912830466077831983260296060, −2.33536529370592371480742998371, −1.30551341946209791454788111753, 0,
1.30551341946209791454788111753, 2.33536529370592371480742998371, 3.43912830466077831983260296060, 4.25317810266192389378879969849, 4.97069578950949997794214798605, 5.94332397504937360956820682604, 6.29211351940374144460927791725, 7.12849542224079283196468796051, 8.041459520984952611250563019274
