| L(s) = 1 | + 2.80·2-s + 5.84·4-s + 3.48·5-s − 1.41·7-s + 10.7·8-s + 9.74·10-s + 4.43·11-s − 2.37·13-s − 3.95·14-s + 18.4·16-s + 3.28·17-s − 3.46·19-s + 20.3·20-s + 12.4·22-s − 23-s + 7.11·25-s − 6.65·26-s − 8.26·28-s − 29-s − 4.18·31-s + 30.1·32-s + 9.20·34-s − 4.92·35-s − 10.7·37-s − 9.69·38-s + 37.4·40-s − 3.71·41-s + ⋯ |
| L(s) = 1 | + 1.98·2-s + 2.92·4-s + 1.55·5-s − 0.534·7-s + 3.80·8-s + 3.08·10-s + 1.33·11-s − 0.659·13-s − 1.05·14-s + 4.61·16-s + 0.797·17-s − 0.794·19-s + 4.54·20-s + 2.64·22-s − 0.208·23-s + 1.42·25-s − 1.30·26-s − 1.56·28-s − 0.185·29-s − 0.751·31-s + 5.33·32-s + 1.57·34-s − 0.831·35-s − 1.76·37-s − 1.57·38-s + 5.92·40-s − 0.579·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.67543777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.67543777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.80T + 2T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 - 4.43T + 11T^{2} \) |
| 13 | \( 1 + 2.37T + 13T^{2} \) |
| 17 | \( 1 - 3.28T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 31 | \( 1 + 4.18T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 3.71T + 41T^{2} \) |
| 43 | \( 1 + 7.08T + 43T^{2} \) |
| 47 | \( 1 + 1.63T + 47T^{2} \) |
| 53 | \( 1 + 6.97T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 5.25T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 8.68T + 73T^{2} \) |
| 79 | \( 1 - 9.51T + 79T^{2} \) |
| 83 | \( 1 + 2.12T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−7.65191719379900042668351129771, −6.82939056385690754808046233283, −6.40341510762940776586380903617, −5.92284903671400282391849575642, −5.18497169560410069490756193110, −4.60995892188870787990069373356, −3.55972757868341912636647917234, −3.10968202198324001436578977984, −1.93529755383164267312860051313, −1.63929170077766445747920240503,
1.63929170077766445747920240503, 1.93529755383164267312860051313, 3.10968202198324001436578977984, 3.55972757868341912636647917234, 4.60995892188870787990069373356, 5.18497169560410069490756193110, 5.92284903671400282391849575642, 6.40341510762940776586380903617, 6.82939056385690754808046233283, 7.65191719379900042668351129771
