| L(s) = 1 | + 3-s + 5-s − 5·7-s − 2·9-s + 6·11-s + 15-s − 5·17-s + 2·19-s − 5·21-s − 2·23-s − 4·25-s − 5·27-s − 8·31-s + 6·33-s − 5·35-s + 5·37-s + 2·41-s + 3·43-s − 2·45-s − 3·47-s + 18·49-s − 5·51-s − 10·53-s + 6·55-s + 2·57-s − 8·59-s − 10·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.88·7-s − 2/3·9-s + 1.80·11-s + 0.258·15-s − 1.21·17-s + 0.458·19-s − 1.09·21-s − 0.417·23-s − 4/5·25-s − 0.962·27-s − 1.43·31-s + 1.04·33-s − 0.845·35-s + 0.821·37-s + 0.312·41-s + 0.457·43-s − 0.298·45-s − 0.437·47-s + 18/7·49-s − 0.700·51-s − 1.37·53-s + 0.809·55-s + 0.264·57-s − 1.04·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 305552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 305552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 113 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.80674114328362, −12.65188664536568, −12.01440089700235, −11.51794425867522, −11.14606933181821, −10.61177757250586, −9.850471053444727, −9.527069408031043, −9.316473559617468, −8.934842565175423, −8.534929822430574, −7.647517247621817, −7.387974149107996, −6.542894121728817, −6.456483688865431, −5.949023522490820, −5.632380962648366, −4.654924367720763, −4.136477765540993, −3.582720176949889, −3.319113778926491, −2.724740843260852, −2.072803796353547, −1.635517903280001, −0.6502545822936474, 0,
0.6502545822936474, 1.635517903280001, 2.072803796353547, 2.724740843260852, 3.319113778926491, 3.582720176949889, 4.136477765540993, 4.654924367720763, 5.632380962648366, 5.949023522490820, 6.456483688865431, 6.542894121728817, 7.387974149107996, 7.647517247621817, 8.534929822430574, 8.934842565175423, 9.316473559617468, 9.527069408031043, 9.850471053444727, 10.61177757250586, 11.14606933181821, 11.51794425867522, 12.01440089700235, 12.65188664536568, 12.80674114328362
