| L(s) = 1 | − 1.53·2-s + 0.364·4-s + 2.97·5-s + 1.52·7-s + 2.51·8-s − 4.56·10-s − 0.657·11-s + 3.12·13-s − 2.34·14-s − 4.59·16-s − 1.99·17-s + 6.50·19-s + 1.08·20-s + 1.01·22-s + 23-s + 3.82·25-s − 4.80·26-s + 0.554·28-s + 29-s + 6.58·31-s + 2.03·32-s + 3.06·34-s + 4.52·35-s + 7.52·37-s − 10.0·38-s + 7.47·40-s − 8.68·41-s + ⋯ |
| L(s) = 1 | − 1.08·2-s + 0.182·4-s + 1.32·5-s + 0.575·7-s + 0.889·8-s − 1.44·10-s − 0.198·11-s + 0.866·13-s − 0.625·14-s − 1.14·16-s − 0.483·17-s + 1.49·19-s + 0.241·20-s + 0.215·22-s + 0.208·23-s + 0.765·25-s − 0.942·26-s + 0.104·28-s + 0.185·29-s + 1.18·31-s + 0.359·32-s + 0.525·34-s + 0.764·35-s + 1.23·37-s − 1.62·38-s + 1.18·40-s − 1.35·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.680486251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.680486251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 5 | \( 1 - 2.97T + 5T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 + 0.657T + 11T^{2} \) |
| 13 | \( 1 - 3.12T + 13T^{2} \) |
| 17 | \( 1 + 1.99T + 17T^{2} \) |
| 19 | \( 1 - 6.50T + 19T^{2} \) |
| 31 | \( 1 - 6.58T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 + 8.68T + 41T^{2} \) |
| 43 | \( 1 + 3.14T + 43T^{2} \) |
| 47 | \( 1 + 1.88T + 47T^{2} \) |
| 53 | \( 1 - 2.78T + 53T^{2} \) |
| 59 | \( 1 + 6.26T + 59T^{2} \) |
| 61 | \( 1 - 1.61T + 61T^{2} \) |
| 67 | \( 1 - 5.54T + 67T^{2} \) |
| 71 | \( 1 - 9.71T + 71T^{2} \) |
| 73 | \( 1 - 4.17T + 73T^{2} \) |
| 79 | \( 1 - 9.99T + 79T^{2} \) |
| 83 | \( 1 + 3.81T + 83T^{2} \) |
| 89 | \( 1 - 0.899T + 89T^{2} \) |
| 97 | \( 1 + 3.81T + 97T^{2} \) |
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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
−8.239322307393786285668008367969, −7.61320271915131226116975261718, −6.70502828007317337815270385810, −6.09928418019640858749425644055, −5.16049474965838324772088654459, −4.72366469592589568045002594618, −3.50823681209218402931658677473, −2.41310135911201090805862727785, −1.57878090744203103526579059495, −0.890283358461757745656255732122,
0.890283358461757745656255732122, 1.57878090744203103526579059495, 2.41310135911201090805862727785, 3.50823681209218402931658677473, 4.72366469592589568045002594618, 5.16049474965838324772088654459, 6.09928418019640858749425644055, 6.70502828007317337815270385810, 7.61320271915131226116975261718, 8.239322307393786285668008367969
