L(s) = 1 | + 0.301·2-s − 1.90·4-s + 3.64·5-s + 7-s − 1.17·8-s + 1.09·10-s + 2.79·11-s + 5.84·13-s + 0.301·14-s + 3.46·16-s − 5.05·17-s + 5.17·19-s − 6.95·20-s + 0.842·22-s − 7.69·23-s + 8.28·25-s + 1.76·26-s − 1.90·28-s + 5.38·29-s + 8.19·31-s + 3.40·32-s − 1.52·34-s + 3.64·35-s − 10.6·37-s + 1.56·38-s − 4.29·40-s − 7.27·41-s + ⋯ |
L(s) = 1 | + 0.213·2-s − 0.954·4-s + 1.62·5-s + 0.377·7-s − 0.416·8-s + 0.347·10-s + 0.843·11-s + 1.62·13-s + 0.0805·14-s + 0.865·16-s − 1.22·17-s + 1.18·19-s − 1.55·20-s + 0.179·22-s − 1.60·23-s + 1.65·25-s + 0.345·26-s − 0.360·28-s + 1.00·29-s + 1.47·31-s + 0.601·32-s − 0.261·34-s + 0.616·35-s − 1.75·37-s + 0.253·38-s − 0.679·40-s − 1.13·41-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
\(L(1)\) |
\(\approx\) |
\(3.189601161\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.189601161\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.301T + 2T^{2} \) |
| 5 | \( 1 - 3.64T + 5T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 - 5.84T + 13T^{2} \) |
| 17 | \( 1 + 5.05T + 17T^{2} \) |
| 19 | \( 1 - 5.17T + 19T^{2} \) |
| 23 | \( 1 + 7.69T + 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 - 8.19T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 7.27T + 41T^{2} \) |
| 43 | \( 1 - 1.97T + 43T^{2} \) |
| 47 | \( 1 - 7.60T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 0.730T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 0.0983T + 71T^{2} \) |
| 73 | \( 1 - 16.7T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 4.72T + 89T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−8.148997955825595787577534387356, −6.78371274413826085051377087405, −6.39652358353833385200723472437, −5.64274452524635088315237532706, −5.21714226181206086553546842956, −4.24107881510305368631199407843, −3.71857497795393731110813559197, −2.62227724612458622629752902959, −1.64785118448733668166058052561, −0.952023023792137218840509380822,
0.952023023792137218840509380822, 1.64785118448733668166058052561, 2.62227724612458622629752902959, 3.71857497795393731110813559197, 4.24107881510305368631199407843, 5.21714226181206086553546842956, 5.64274452524635088315237532706, 6.39652358353833385200723472437, 6.78371274413826085051377087405, 8.148997955825595787577534387356