| L(s) = 1 | + 2.17·2-s + 2.71·4-s + 4.06·5-s + 7-s + 1.55·8-s + 8.83·10-s − 0.266·11-s − 1.09·13-s + 2.17·14-s − 2.05·16-s − 2.89·17-s + 7.39·19-s + 11.0·20-s − 0.579·22-s + 3.52·23-s + 11.5·25-s − 2.37·26-s + 2.71·28-s + 5.16·29-s − 8.51·31-s − 7.57·32-s − 6.28·34-s + 4.06·35-s − 9.79·37-s + 16.0·38-s + 6.31·40-s + 8.53·41-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.35·4-s + 1.81·5-s + 0.377·7-s + 0.548·8-s + 2.79·10-s − 0.0804·11-s − 0.303·13-s + 0.580·14-s − 0.514·16-s − 0.701·17-s + 1.69·19-s + 2.46·20-s − 0.123·22-s + 0.734·23-s + 2.30·25-s − 0.466·26-s + 0.513·28-s + 0.958·29-s − 1.52·31-s − 1.33·32-s − 1.07·34-s + 0.687·35-s − 1.60·37-s + 2.60·38-s + 0.998·40-s + 1.33·41-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
| \(L(1)\) |
\(\approx\) |
\(7.761811051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.761811051\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.17T + 2T^{2} \) |
| 5 | \( 1 - 4.06T + 5T^{2} \) |
| 11 | \( 1 + 0.266T + 11T^{2} \) |
| 13 | \( 1 + 1.09T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 - 7.39T + 19T^{2} \) |
| 23 | \( 1 - 3.52T + 23T^{2} \) |
| 29 | \( 1 - 5.16T + 29T^{2} \) |
| 31 | \( 1 + 8.51T + 31T^{2} \) |
| 37 | \( 1 + 9.79T + 37T^{2} \) |
| 41 | \( 1 - 8.53T + 41T^{2} \) |
| 43 | \( 1 - 6.33T + 43T^{2} \) |
| 47 | \( 1 - 5.55T + 47T^{2} \) |
| 53 | \( 1 - 9.12T + 53T^{2} \) |
| 59 | \( 1 + 0.616T + 59T^{2} \) |
| 61 | \( 1 - 0.338T + 61T^{2} \) |
| 67 | \( 1 + 2.27T + 67T^{2} \) |
| 71 | \( 1 + 4.97T + 71T^{2} \) |
| 73 | \( 1 + 0.185T + 73T^{2} \) |
| 79 | \( 1 + 2.63T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 7.23T + 89T^{2} \) |
| 97 | \( 1 + 1.68T + 97T^{2} \) |
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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
−7.33066343194461397996156052432, −6.99419834016524651000748136738, −6.10783003878086924996462710894, −5.54663846291635972110667494175, −5.19518323663495195305215816847, −4.52442867431386191169433292864, −3.51820452105332205717355813372, −2.68341778977214314337584486696, −2.16887549898752441709160052060, −1.18019459950740819203016857622,
1.18019459950740819203016857622, 2.16887549898752441709160052060, 2.68341778977214314337584486696, 3.51820452105332205717355813372, 4.52442867431386191169433292864, 5.19518323663495195305215816847, 5.54663846291635972110667494175, 6.10783003878086924996462710894, 6.99419834016524651000748136738, 7.33066343194461397996156052432
