| L(s) = 1 | − 2·3-s + 4·5-s − 7-s + 9-s + 4·11-s − 4·13-s − 8·15-s − 17-s + 6·19-s + 2·21-s + 11·25-s + 4·27-s + 6·29-s − 4·31-s − 8·33-s − 4·35-s − 10·37-s + 8·39-s + 6·41-s + 4·45-s − 4·47-s + 49-s + 2·51-s + 14·53-s + 16·55-s − 12·57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.78·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.10·13-s − 2.06·15-s − 0.242·17-s + 1.37·19-s + 0.436·21-s + 11/5·25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s − 1.39·33-s − 0.676·35-s − 1.64·37-s + 1.28·39-s + 0.937·41-s + 0.596·45-s − 0.583·47-s + 1/7·49-s + 0.280·51-s + 1.92·53-s + 2.15·55-s − 1.58·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.601016422\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.601016422\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−9.353877214568608158975685691235, −8.792320134679942386987328204107, −7.20538145889120705419675013150, −6.67715777173991787561748890315, −5.93683910837363468486972715767, −5.38202673164662018917332684771, −4.66926669335912418503034638447, −3.18709894584368791480262864867, −2.06847900519095682932735371240, −0.936906498625533428667018109497,
0.936906498625533428667018109497, 2.06847900519095682932735371240, 3.18709894584368791480262864867, 4.66926669335912418503034638447, 5.38202673164662018917332684771, 5.93683910837363468486972715767, 6.67715777173991787561748890315, 7.20538145889120705419675013150, 8.792320134679942386987328204107, 9.353877214568608158975685691235
