| L(s) = 1 | + 2.57·2-s + 4.62·4-s − 2.78·5-s − 7-s + 6.76·8-s − 7.16·10-s − 1.44·11-s + 4.74·13-s − 2.57·14-s + 8.15·16-s − 4.74·17-s + 5.68·19-s − 12.8·20-s − 3.72·22-s − 0.885·23-s + 2.75·25-s + 12.2·26-s − 4.62·28-s + 1.81·29-s + 4.74·31-s + 7.47·32-s − 12.2·34-s + 2.78·35-s + 5.22·37-s + 14.6·38-s − 18.8·40-s + 5.00·41-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 2.31·4-s − 1.24·5-s − 0.377·7-s + 2.39·8-s − 2.26·10-s − 0.435·11-s + 1.31·13-s − 0.688·14-s + 2.03·16-s − 1.15·17-s + 1.30·19-s − 2.88·20-s − 0.793·22-s − 0.184·23-s + 0.550·25-s + 2.39·26-s − 0.874·28-s + 0.336·29-s + 0.853·31-s + 1.32·32-s − 2.09·34-s + 0.470·35-s + 0.859·37-s + 2.37·38-s − 2.97·40-s + 0.782·41-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
| \(L(1)\) |
\(\approx\) |
\(5.339606825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.339606825\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.57T + 2T^{2} \) |
| 5 | \( 1 + 2.78T + 5T^{2} \) |
| 11 | \( 1 + 1.44T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 - 5.68T + 19T^{2} \) |
| 23 | \( 1 + 0.885T + 23T^{2} \) |
| 29 | \( 1 - 1.81T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 - 5.22T + 37T^{2} \) |
| 41 | \( 1 - 5.00T + 41T^{2} \) |
| 43 | \( 1 + 6.34T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 + 8.73T + 53T^{2} \) |
| 59 | \( 1 - 7.36T + 59T^{2} \) |
| 61 | \( 1 - 2.00T + 61T^{2} \) |
| 67 | \( 1 + 4.76T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 3.77T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 2.64T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.63117761150742005100572558500, −6.90367060888783920518977167750, −6.31712608819720560964130818229, −5.67536195815164453083184791604, −4.84685165127244471892136105736, −4.22333895828900786814647935699, −3.66390773649889199058031991489, −3.07622127854783537886844874262, −2.25286867034000416766973246661, −0.865661118142263179178154242309,
0.865661118142263179178154242309, 2.25286867034000416766973246661, 3.07622127854783537886844874262, 3.66390773649889199058031991489, 4.22333895828900786814647935699, 4.84685165127244471892136105736, 5.67536195815164453083184791604, 6.31712608819720560964130818229, 6.90367060888783920518977167750, 7.63117761150742005100572558500
