| L(s) = 1 | + 2.60·2-s + 4.80·4-s + 1.19·5-s − 4.49·7-s + 7.31·8-s + 3.12·10-s + 5.34·11-s + 2.49·13-s − 11.7·14-s + 9.47·16-s + 3.03·17-s + 2.47·19-s + 5.75·20-s + 13.9·22-s + 23-s − 3.56·25-s + 6.51·26-s − 21.6·28-s − 29-s − 10.2·31-s + 10.0·32-s + 7.91·34-s − 5.38·35-s + 9.08·37-s + 6.45·38-s + 8.76·40-s + 0.132·41-s + ⋯ |
| L(s) = 1 | + 1.84·2-s + 2.40·4-s + 0.535·5-s − 1.69·7-s + 2.58·8-s + 0.987·10-s + 1.61·11-s + 0.693·13-s − 3.13·14-s + 2.36·16-s + 0.735·17-s + 0.567·19-s + 1.28·20-s + 2.97·22-s + 0.208·23-s − 0.713·25-s + 1.27·26-s − 4.08·28-s − 0.185·29-s − 1.84·31-s + 1.78·32-s + 1.35·34-s − 0.910·35-s + 1.49·37-s + 1.04·38-s + 1.38·40-s + 0.0206·41-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
| \(L(1)\) |
\(\approx\) |
\(7.392364691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.392364691\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.19T + 5T^{2} \) |
| 7 | \( 1 + 4.49T + 7T^{2} \) |
| 11 | \( 1 - 5.34T + 11T^{2} \) |
| 13 | \( 1 - 2.49T + 13T^{2} \) |
| 17 | \( 1 - 3.03T + 17T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 9.08T + 37T^{2} \) |
| 41 | \( 1 - 0.132T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 - 6.82T + 47T^{2} \) |
| 53 | \( 1 - 3.33T + 53T^{2} \) |
| 59 | \( 1 - 9.14T + 59T^{2} \) |
| 61 | \( 1 + 7.63T + 61T^{2} \) |
| 67 | \( 1 + 14.0T + 67T^{2} \) |
| 71 | \( 1 - 15.9T + 71T^{2} \) |
| 73 | \( 1 + 1.85T + 73T^{2} \) |
| 79 | \( 1 + 9.29T + 79T^{2} \) |
| 83 | \( 1 - 7.74T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.58860107780834608608003546828, −7.00323617369794669476605035675, −6.28126739000776361698458897648, −5.96266246527084998111462404321, −5.40781913974676556659104215121, −4.16710197884859806600382854669, −3.70245315133760894148713843897, −3.19942901894639616458287864547, −2.23828228970597430166719732066, −1.16575271218898161810466911704,
1.16575271218898161810466911704, 2.23828228970597430166719732066, 3.19942901894639616458287864547, 3.70245315133760894148713843897, 4.16710197884859806600382854669, 5.40781913974676556659104215121, 5.96266246527084998111462404321, 6.28126739000776361698458897648, 7.00323617369794669476605035675, 7.58860107780834608608003546828
