| L(s) = 1 | + 3.07·3-s + 4.23·5-s + 2.35·7-s + 6.47·9-s − 4.53·11-s + 1.76·13-s + 13.0·15-s − 5.23·17-s − 6.15·19-s + 7.23·21-s − 3.80·23-s + 12.9·25-s + 10.6·27-s − 29-s + 0.726·31-s − 13.9·33-s + 9.95·35-s − 2.47·37-s + 5.42·39-s − 7.23·41-s + 5.98·43-s + 27.4·45-s − 5.42·47-s − 1.47·49-s − 16.1·51-s + 3.76·53-s − 19.1·55-s + ⋯ |
| L(s) = 1 | + 1.77·3-s + 1.89·5-s + 0.888·7-s + 2.15·9-s − 1.36·11-s + 0.489·13-s + 3.36·15-s − 1.26·17-s − 1.41·19-s + 1.57·21-s − 0.793·23-s + 2.58·25-s + 2.05·27-s − 0.185·29-s + 0.130·31-s − 2.42·33-s + 1.68·35-s − 0.406·37-s + 0.869·39-s − 1.13·41-s + 0.912·43-s + 4.08·45-s − 0.791·47-s − 0.210·49-s − 2.25·51-s + 0.517·53-s − 2.58·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.537464628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.537464628\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 3.07T + 3T^{2} \) |
| 5 | \( 1 - 4.23T + 5T^{2} \) |
| 7 | \( 1 - 2.35T + 7T^{2} \) |
| 11 | \( 1 + 4.53T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 6.15T + 19T^{2} \) |
| 23 | \( 1 + 3.80T + 23T^{2} \) |
| 31 | \( 1 - 0.726T + 31T^{2} \) |
| 37 | \( 1 + 2.47T + 37T^{2} \) |
| 41 | \( 1 + 7.23T + 41T^{2} \) |
| 43 | \( 1 - 5.98T + 43T^{2} \) |
| 47 | \( 1 + 5.42T + 47T^{2} \) |
| 53 | \( 1 - 3.76T + 53T^{2} \) |
| 59 | \( 1 + 6.71T + 59T^{2} \) |
| 61 | \( 1 - 2.76T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 3.24T + 71T^{2} \) |
| 73 | \( 1 + 4.94T + 73T^{2} \) |
| 79 | \( 1 + 2.17T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 - 1.70T + 89T^{2} \) |
| 97 | \( 1 - 9.70T + 97T^{2} \) |
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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
−9.012525872001355791187058543256, −8.583929495155811365636575464257, −8.000231329588973762111762904896, −6.95030329851201774129363409046, −6.10341019536012158334265609982, −5.07776982945170524851662688507, −4.27602492103282518911972063278, −2.93852573348348348886598796004, −2.09870660412694756422059348330, −1.82769237587888095358949511676,
1.82769237587888095358949511676, 2.09870660412694756422059348330, 2.93852573348348348886598796004, 4.27602492103282518911972063278, 5.07776982945170524851662688507, 6.10341019536012158334265609982, 6.95030329851201774129363409046, 8.000231329588973762111762904896, 8.583929495155811365636575464257, 9.012525872001355791187058543256
