| L(s) = 1 | + 2·3-s + 5-s + 2·7-s + 9-s + 4·11-s + 2·13-s + 2·15-s − 4·17-s + 4·19-s + 4·21-s + 25-s − 4·27-s − 6·29-s + 8·31-s + 8·33-s + 2·35-s + 8·37-s + 4·39-s − 2·41-s + 10·43-s + 45-s − 2·47-s − 3·49-s − 8·51-s − 12·53-s + 4·55-s + 8·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s − 0.970·17-s + 0.917·19-s + 0.872·21-s + 1/5·25-s − 0.769·27-s − 1.11·29-s + 1.43·31-s + 1.39·33-s + 0.338·35-s + 1.31·37-s + 0.640·39-s − 0.312·41-s + 1.52·43-s + 0.149·45-s − 0.291·47-s − 3/7·49-s − 1.12·51-s − 1.64·53-s + 0.539·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.205895942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.205895942\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−8.040721370974225166210857412146, −7.60131094092934614062007472306, −6.57940715487016495444241943251, −6.07168116492758495052613678127, −5.07082329034147701370815702415, −4.25810016007178759574644517759, −3.59297703667070069965696659492, −2.70259536619826786269143930281, −1.92054604388890799191821650783, −1.09756009964482636156470610850,
1.09756009964482636156470610850, 1.92054604388890799191821650783, 2.70259536619826786269143930281, 3.59297703667070069965696659492, 4.25810016007178759574644517759, 5.07082329034147701370815702415, 6.07168116492758495052613678127, 6.57940715487016495444241943251, 7.60131094092934614062007472306, 8.040721370974225166210857412146
