L(s) = 1 | + 1.42·2-s + 0.0348·4-s − 4.07·5-s + 7-s − 2.80·8-s − 5.81·10-s − 3.50·11-s − 5.98·13-s + 1.42·14-s − 4.06·16-s − 7.04·17-s + 3.99·19-s − 0.142·20-s − 4.99·22-s − 7.32·23-s + 11.6·25-s − 8.53·26-s + 0.0348·28-s − 7.77·29-s + 2.84·31-s − 0.197·32-s − 10.0·34-s − 4.07·35-s − 1.23·37-s + 5.70·38-s + 11.4·40-s − 1.75·41-s + ⋯ |
L(s) = 1 | + 1.00·2-s + 0.0174·4-s − 1.82·5-s + 0.377·7-s − 0.991·8-s − 1.83·10-s − 1.05·11-s − 1.65·13-s + 0.381·14-s − 1.01·16-s − 1.70·17-s + 0.917·19-s − 0.0318·20-s − 1.06·22-s − 1.52·23-s + 2.32·25-s − 1.67·26-s + 0.00659·28-s − 1.44·29-s + 0.510·31-s − 0.0348·32-s − 1.72·34-s − 0.689·35-s − 0.202·37-s + 0.925·38-s + 1.80·40-s − 0.274·41-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
\(L(1)\) |
\(\approx\) |
\(0.008726446836\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.008726446836\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.42T + 2T^{2} \) |
| 5 | \( 1 + 4.07T + 5T^{2} \) |
| 11 | \( 1 + 3.50T + 11T^{2} \) |
| 13 | \( 1 + 5.98T + 13T^{2} \) |
| 17 | \( 1 + 7.04T + 17T^{2} \) |
| 19 | \( 1 - 3.99T + 19T^{2} \) |
| 23 | \( 1 + 7.32T + 23T^{2} \) |
| 29 | \( 1 + 7.77T + 29T^{2} \) |
| 31 | \( 1 - 2.84T + 31T^{2} \) |
| 37 | \( 1 + 1.23T + 37T^{2} \) |
| 41 | \( 1 + 1.75T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 4.77T + 47T^{2} \) |
| 53 | \( 1 - 4.77T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 2.14T + 61T^{2} \) |
| 67 | \( 1 + 5.59T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 5.10T + 73T^{2} \) |
| 79 | \( 1 - 8.07T + 79T^{2} \) |
| 83 | \( 1 - 5.09T + 83T^{2} \) |
| 89 | \( 1 + 0.679T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−7.87612806636030688845934082918, −7.17511603130160938690311574540, −6.49116763865047515895178445275, −5.32009214632513175255703988226, −4.91083029559909931337297146697, −4.35987755804841071413730803410, −3.71198579186851536089002988750, −2.94333632376135452135371649599, −2.14071888337167092366438786528, −0.03645257865752169123783842547,
0.03645257865752169123783842547, 2.14071888337167092366438786528, 2.94333632376135452135371649599, 3.71198579186851536089002988750, 4.35987755804841071413730803410, 4.91083029559909931337297146697, 5.32009214632513175255703988226, 6.49116763865047515895178445275, 7.17511603130160938690311574540, 7.87612806636030688845934082918