| L(s) = 1 | + 2.17·2-s − 3-s + 2.73·4-s − 5-s − 2.17·6-s − 0.443·7-s + 1.59·8-s + 9-s − 2.17·10-s + 1.59·11-s − 2.73·12-s − 0.964·14-s + 15-s − 1.99·16-s + 3.76·17-s + 2.17·18-s + 1.54·19-s − 2.73·20-s + 0.443·21-s + 3.46·22-s + 0.705·23-s − 1.59·24-s + 25-s − 27-s − 1.21·28-s + 2.49·29-s + 2.17·30-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 0.577·3-s + 1.36·4-s − 0.447·5-s − 0.888·6-s − 0.167·7-s + 0.563·8-s + 0.333·9-s − 0.687·10-s + 0.480·11-s − 0.788·12-s − 0.257·14-s + 0.258·15-s − 0.499·16-s + 0.913·17-s + 0.512·18-s + 0.354·19-s − 0.610·20-s + 0.0967·21-s + 0.738·22-s + 0.147·23-s − 0.325·24-s + 0.200·25-s − 0.192·27-s − 0.228·28-s + 0.464·29-s + 0.397·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.364847383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.364847383\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 7 | \( 1 + 0.443T + 7T^{2} \) |
| 11 | \( 1 - 1.59T + 11T^{2} \) |
| 17 | \( 1 - 3.76T + 17T^{2} \) |
| 19 | \( 1 - 1.54T + 19T^{2} \) |
| 23 | \( 1 - 0.705T + 23T^{2} \) |
| 29 | \( 1 - 2.49T + 29T^{2} \) |
| 31 | \( 1 - 6.18T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 - 6.02T + 41T^{2} \) |
| 43 | \( 1 - 4.72T + 43T^{2} \) |
| 47 | \( 1 - 5.41T + 47T^{2} \) |
| 53 | \( 1 - 5.51T + 53T^{2} \) |
| 59 | \( 1 - 8.72T + 59T^{2} \) |
| 61 | \( 1 + 9.73T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 9.12T + 83T^{2} \) |
| 89 | \( 1 - 4.75T + 89T^{2} \) |
| 97 | \( 1 - 8.41T + 97T^{2} \) |
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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
−8.936121723089860217318773170956, −7.80445072421324979278110006792, −7.14264917217740755200061555961, −6.18586784260321296075423182302, −5.84277852529084063988665165202, −4.80111232549878963614820562757, −4.28648467496177210343755690342, −3.41376348200194829867870578848, −2.57868157691315766289305029873, −0.984903352843816885537772188385,
0.984903352843816885537772188385, 2.57868157691315766289305029873, 3.41376348200194829867870578848, 4.28648467496177210343755690342, 4.80111232549878963614820562757, 5.84277852529084063988665165202, 6.18586784260321296075423182302, 7.14264917217740755200061555961, 7.80445072421324979278110006792, 8.936121723089860217318773170956
