| L(s) = 1 | + 2.60·2-s + 4.80·4-s − 3.69·5-s + 1.51·7-s + 7.32·8-s − 9.63·10-s − 4.23·11-s + 0.701·13-s + 3.94·14-s + 9.49·16-s − 2.20·17-s + 3.51·19-s − 17.7·20-s − 11.0·22-s + 23-s + 8.62·25-s + 1.82·26-s + 7.27·28-s − 29-s + 8.97·31-s + 10.1·32-s − 5.76·34-s − 5.58·35-s + 9.90·37-s + 9.17·38-s − 27.0·40-s + 8.88·41-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 2.40·4-s − 1.65·5-s + 0.571·7-s + 2.58·8-s − 3.04·10-s − 1.27·11-s + 0.194·13-s + 1.05·14-s + 2.37·16-s − 0.535·17-s + 0.806·19-s − 3.96·20-s − 2.35·22-s + 0.208·23-s + 1.72·25-s + 0.358·26-s + 1.37·28-s − 0.185·29-s + 1.61·31-s + 1.78·32-s − 0.988·34-s − 0.943·35-s + 1.62·37-s + 1.48·38-s − 4.27·40-s + 1.38·41-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
| \(L(1)\) |
\(\approx\) |
\(5.148660625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.148660625\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.60T + 2T^{2} \) |
| 5 | \( 1 + 3.69T + 5T^{2} \) |
| 7 | \( 1 - 1.51T + 7T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 - 0.701T + 13T^{2} \) |
| 17 | \( 1 + 2.20T + 17T^{2} \) |
| 19 | \( 1 - 3.51T + 19T^{2} \) |
| 31 | \( 1 - 8.97T + 31T^{2} \) |
| 37 | \( 1 - 9.90T + 37T^{2} \) |
| 41 | \( 1 - 8.88T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 9.05T + 47T^{2} \) |
| 53 | \( 1 + 4.47T + 53T^{2} \) |
| 59 | \( 1 - 8.07T + 59T^{2} \) |
| 61 | \( 1 + 0.602T + 61T^{2} \) |
| 67 | \( 1 - 2.51T + 67T^{2} \) |
| 71 | \( 1 - 4.90T + 71T^{2} \) |
| 73 | \( 1 + 0.365T + 73T^{2} \) |
| 79 | \( 1 + 1.39T + 79T^{2} \) |
| 83 | \( 1 + 0.338T + 83T^{2} \) |
| 89 | \( 1 - 6.33T + 89T^{2} \) |
| 97 | \( 1 + 5.60T + 97T^{2} \) |
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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
−7.78155108702880751458256507699, −7.39017012083911798374941429798, −6.53631568096875499864091202703, −5.67735145357194524174355158395, −5.01500218079900774234693711397, −4.31256508993498644949338110542, −3.99780874373516821863660564897, −2.88982460758743811821064655535, −2.52979718342073088540542811272, −0.905334761144000343456257760610,
0.905334761144000343456257760610, 2.52979718342073088540542811272, 2.88982460758743811821064655535, 3.99780874373516821863660564897, 4.31256508993498644949338110542, 5.01500218079900774234693711397, 5.67735145357194524174355158395, 6.53631568096875499864091202703, 7.39017012083911798374941429798, 7.78155108702880751458256507699
