| L(s) = 1 | + 2-s + 3-s + 4-s + 0.888·5-s + 6-s + 4.32·7-s + 8-s + 9-s + 0.888·10-s − 3.02·11-s + 12-s − 3.43·13-s + 4.32·14-s + 0.888·15-s + 16-s + 7.91·17-s + 18-s + 0.799·19-s + 0.888·20-s + 4.32·21-s − 3.02·22-s + 23-s + 24-s − 4.21·25-s − 3.43·26-s + 27-s + 4.32·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.397·5-s + 0.408·6-s + 1.63·7-s + 0.353·8-s + 0.333·9-s + 0.280·10-s − 0.912·11-s + 0.288·12-s − 0.953·13-s + 1.15·14-s + 0.229·15-s + 0.250·16-s + 1.91·17-s + 0.235·18-s + 0.183·19-s + 0.198·20-s + 0.943·21-s − 0.645·22-s + 0.208·23-s + 0.204·24-s − 0.842·25-s − 0.673·26-s + 0.192·27-s + 0.817·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.929524806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.929524806\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 0.888T + 5T^{2} \) |
| 7 | \( 1 - 4.32T + 7T^{2} \) |
| 11 | \( 1 + 3.02T + 11T^{2} \) |
| 13 | \( 1 + 3.43T + 13T^{2} \) |
| 17 | \( 1 - 7.91T + 17T^{2} \) |
| 19 | \( 1 - 0.799T + 19T^{2} \) |
| 31 | \( 1 + 6.85T + 31T^{2} \) |
| 37 | \( 1 - 9.31T + 37T^{2} \) |
| 41 | \( 1 - 9.48T + 41T^{2} \) |
| 43 | \( 1 - 8.81T + 43T^{2} \) |
| 47 | \( 1 + 7.39T + 47T^{2} \) |
| 53 | \( 1 + 1.38T + 53T^{2} \) |
| 59 | \( 1 + 7.21T + 59T^{2} \) |
| 61 | \( 1 - 6.80T + 61T^{2} \) |
| 67 | \( 1 - 7.98T + 67T^{2} \) |
| 71 | \( 1 + 5.25T + 71T^{2} \) |
| 73 | \( 1 + 7.90T + 73T^{2} \) |
| 79 | \( 1 + 8.50T + 79T^{2} \) |
| 83 | \( 1 - 5.93T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 9.12T + 97T^{2} \) |
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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
−8.018890511386373017741488567921, −7.76918548291419815602554981374, −7.30679551861759008148370002208, −5.86937312954415467888134111736, −5.41090705815870647641840631673, −4.74694656587831076498965668898, −3.93448209730127599560963759469, −2.85038551981881069494919679555, −2.17649386024949516564871134621, −1.24135374280326048937371532089,
1.24135374280326048937371532089, 2.17649386024949516564871134621, 2.85038551981881069494919679555, 3.93448209730127599560963759469, 4.74694656587831076498965668898, 5.41090705815870647641840631673, 5.86937312954415467888134111736, 7.30679551861759008148370002208, 7.76918548291419815602554981374, 8.018890511386373017741488567921
