| L(s) = 1 | + 2-s − 4-s − 2·5-s + 3·7-s − 3·8-s − 2·10-s + 3·11-s − 6·13-s + 3·14-s − 16-s + 5·17-s − 2·19-s + 2·20-s + 3·22-s − 25-s − 6·26-s − 3·28-s + 7·29-s + 5·31-s + 5·32-s + 5·34-s − 6·35-s + 11·37-s − 2·38-s + 6·40-s + 2·41-s + 8·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s + 1.13·7-s − 1.06·8-s − 0.632·10-s + 0.904·11-s − 1.66·13-s + 0.801·14-s − 1/4·16-s + 1.21·17-s − 0.458·19-s + 0.447·20-s + 0.639·22-s − 1/5·25-s − 1.17·26-s − 0.566·28-s + 1.29·29-s + 0.898·31-s + 0.883·32-s + 0.857·34-s − 1.01·35-s + 1.80·37-s − 0.324·38-s + 0.948·40-s + 0.312·41-s + 1.21·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 395163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395163 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 23 | \( 1 \) | |
| 83 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−12.44750489798068, −12.27899456771404, −11.88810873759142, −11.47901807963228, −11.17230171351844, −10.32423666152785, −9.964831627181977, −9.574511065324727, −8.980989365019054, −8.517808141169980, −8.085594570083090, −7.569728376919471, −7.422941974919058, −6.628034601798458, −5.999599652804741, −5.739147389128465, −4.885119890008801, −4.685245073946106, −4.344769813222043, −3.891008573511400, −3.239876345830585, −2.678342069386550, −2.228131591413439, −1.199354323825440, −0.8371893525000941, 0,
0.8371893525000941, 1.199354323825440, 2.228131591413439, 2.678342069386550, 3.239876345830585, 3.891008573511400, 4.344769813222043, 4.685245073946106, 4.885119890008801, 5.739147389128465, 5.999599652804741, 6.628034601798458, 7.422941974919058, 7.569728376919471, 8.085594570083090, 8.517808141169980, 8.980989365019054, 9.574511065324727, 9.964831627181977, 10.32423666152785, 11.17230171351844, 11.47901807963228, 11.88810873759142, 12.27899456771404, 12.44750489798068
