| L(s) = 1 | + 1.20·2-s − 0.559·4-s − 3.40·5-s + 4.71·7-s − 3.07·8-s − 4.08·10-s − 1.68·11-s − 2.33·13-s + 5.65·14-s − 2.56·16-s − 4.01·17-s + 0.666·19-s + 1.90·20-s − 2.02·22-s + 23-s + 6.56·25-s − 2.80·26-s − 2.63·28-s − 29-s − 4.38·31-s + 3.06·32-s − 4.82·34-s − 16.0·35-s + 1.15·37-s + 0.799·38-s + 10.4·40-s + 10.3·41-s + ⋯ |
| L(s) = 1 | + 0.848·2-s − 0.279·4-s − 1.52·5-s + 1.78·7-s − 1.08·8-s − 1.29·10-s − 0.508·11-s − 0.647·13-s + 1.51·14-s − 0.641·16-s − 0.974·17-s + 0.152·19-s + 0.425·20-s − 0.431·22-s + 0.208·23-s + 1.31·25-s − 0.549·26-s − 0.498·28-s − 0.185·29-s − 0.787·31-s + 0.541·32-s − 0.826·34-s − 2.70·35-s + 0.189·37-s + 0.129·38-s + 1.65·40-s + 1.61·41-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
| \(L(1)\) |
\(\approx\) |
\(1.578745607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.578745607\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.20T + 2T^{2} \) |
| 5 | \( 1 + 3.40T + 5T^{2} \) |
| 7 | \( 1 - 4.71T + 7T^{2} \) |
| 11 | \( 1 + 1.68T + 11T^{2} \) |
| 13 | \( 1 + 2.33T + 13T^{2} \) |
| 17 | \( 1 + 4.01T + 17T^{2} \) |
| 19 | \( 1 - 0.666T + 19T^{2} \) |
| 31 | \( 1 + 4.38T + 31T^{2} \) |
| 37 | \( 1 - 1.15T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4.96T + 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 7.60T + 53T^{2} \) |
| 59 | \( 1 - 2.67T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 7.64T + 67T^{2} \) |
| 71 | \( 1 + 2.23T + 71T^{2} \) |
| 73 | \( 1 - 3.18T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 0.691T + 83T^{2} \) |
| 89 | \( 1 + 18.4T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.110870135377252936061413338227, −7.45130169411014004930775345061, −6.77743455804314239140193564561, −5.49453220840274833546191576330, −5.06814997518461984435356004519, −4.35725499149527922187122841362, −4.03304420119623697197648426489, −3.00542440409421802178023011084, −2.04266180806912475029017908243, −0.56911699022471455224205087135,
0.56911699022471455224205087135, 2.04266180806912475029017908243, 3.00542440409421802178023011084, 4.03304420119623697197648426489, 4.35725499149527922187122841362, 5.06814997518461984435356004519, 5.49453220840274833546191576330, 6.77743455804314239140193564561, 7.45130169411014004930775345061, 8.110870135377252936061413338227
