| L(s) = 1 | + 2.70·2-s + 5.31·4-s + 1.33·5-s − 7-s + 8.96·8-s + 3.61·10-s + 5.62·11-s + 3.64·13-s − 2.70·14-s + 13.6·16-s + 6.28·17-s + 2.90·19-s + 7.10·20-s + 15.2·22-s − 5.83·23-s − 3.21·25-s + 9.84·26-s − 5.31·28-s + 4.25·29-s − 9.19·31-s + 18.8·32-s + 17.0·34-s − 1.33·35-s − 7.71·37-s + 7.84·38-s + 11.9·40-s − 2.94·41-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 2.65·4-s + 0.597·5-s − 0.377·7-s + 3.16·8-s + 1.14·10-s + 1.69·11-s + 1.00·13-s − 0.722·14-s + 3.40·16-s + 1.52·17-s + 0.665·19-s + 1.58·20-s + 3.24·22-s − 1.21·23-s − 0.642·25-s + 1.93·26-s − 1.00·28-s + 0.790·29-s − 1.65·31-s + 3.33·32-s + 2.91·34-s − 0.226·35-s − 1.26·37-s + 1.27·38-s + 1.89·40-s − 0.460·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.08761957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.08761957\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 5 | \( 1 - 1.33T + 5T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 - 6.28T + 17T^{2} \) |
| 19 | \( 1 - 2.90T + 19T^{2} \) |
| 23 | \( 1 + 5.83T + 23T^{2} \) |
| 29 | \( 1 - 4.25T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 + 7.71T + 37T^{2} \) |
| 41 | \( 1 + 2.94T + 41T^{2} \) |
| 43 | \( 1 + 9.41T + 43T^{2} \) |
| 47 | \( 1 + 1.87T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 5.75T + 59T^{2} \) |
| 61 | \( 1 - 0.977T + 61T^{2} \) |
| 67 | \( 1 + 0.352T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 0.00843T + 79T^{2} \) |
| 83 | \( 1 + 0.0999T + 83T^{2} \) |
| 89 | \( 1 + 8.26T + 89T^{2} \) |
| 97 | \( 1 - 3.01T + 97T^{2} \) |
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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
−7.49311534599940705236277652553, −6.69676766757192986414993319264, −6.26190014169028729753878342591, −5.71726208748700930700886479727, −5.14382071422006366371575576219, −4.14438570906530363308401123540, −3.48782817831982350749337001346, −3.25846018521438141231768007923, −1.80318162523068916145573540255, −1.45065553114219909094905788044,
1.45065553114219909094905788044, 1.80318162523068916145573540255, 3.25846018521438141231768007923, 3.48782817831982350749337001346, 4.14438570906530363308401123540, 5.14382071422006366371575576219, 5.71726208748700930700886479727, 6.26190014169028729753878342591, 6.69676766757192986414993319264, 7.49311534599940705236277652553
