| L(s) = 1 | + 3-s + 0.649·5-s + 4.19·7-s + 9-s + 1.95·11-s + 3.07·13-s + 0.649·15-s + 6.85·17-s + 8.04·19-s + 4.19·21-s − 6.50·23-s − 4.57·25-s + 27-s − 2.52·29-s + 4.14·31-s + 1.95·33-s + 2.72·35-s − 4.94·37-s + 3.07·39-s + 2.05·41-s + 4.35·43-s + 0.649·45-s − 11.2·47-s + 10.6·49-s + 6.85·51-s − 1.69·53-s + 1.26·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.290·5-s + 1.58·7-s + 0.333·9-s + 0.588·11-s + 0.852·13-s + 0.167·15-s + 1.66·17-s + 1.84·19-s + 0.915·21-s − 1.35·23-s − 0.915·25-s + 0.192·27-s − 0.468·29-s + 0.745·31-s + 0.339·33-s + 0.460·35-s − 0.813·37-s + 0.492·39-s + 0.321·41-s + 0.664·43-s + 0.0967·45-s − 1.63·47-s + 1.51·49-s + 0.959·51-s − 0.233·53-s + 0.170·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.679352448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.679352448\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 - 0.649T + 5T^{2} \) |
| 7 | \( 1 - 4.19T + 7T^{2} \) |
| 11 | \( 1 - 1.95T + 11T^{2} \) |
| 13 | \( 1 - 3.07T + 13T^{2} \) |
| 17 | \( 1 - 6.85T + 17T^{2} \) |
| 19 | \( 1 - 8.04T + 19T^{2} \) |
| 23 | \( 1 + 6.50T + 23T^{2} \) |
| 29 | \( 1 + 2.52T + 29T^{2} \) |
| 31 | \( 1 - 4.14T + 31T^{2} \) |
| 37 | \( 1 + 4.94T + 37T^{2} \) |
| 41 | \( 1 - 2.05T + 41T^{2} \) |
| 43 | \( 1 - 4.35T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 1.69T + 53T^{2} \) |
| 59 | \( 1 + 7.31T + 59T^{2} \) |
| 61 | \( 1 + 12.5T + 61T^{2} \) |
| 67 | \( 1 + 6.97T + 67T^{2} \) |
| 71 | \( 1 - 2.84T + 71T^{2} \) |
| 73 | \( 1 + 7.78T + 73T^{2} \) |
| 79 | \( 1 + 3.29T + 79T^{2} \) |
| 83 | \( 1 - 1.99T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−8.181609956392185039043808536273, −7.914242853678537265424321248442, −7.28762899546426866318493413724, −6.03900947776268289320334604685, −5.53668798868602356028454055948, −4.64101587430246861042639677780, −3.78534667177610382795374806566, −3.04057021772645848054473286397, −1.66258942351646952689952137171, −1.32068022978118988518037368326,
1.32068022978118988518037368326, 1.66258942351646952689952137171, 3.04057021772645848054473286397, 3.78534667177610382795374806566, 4.64101587430246861042639677780, 5.53668798868602356028454055948, 6.03900947776268289320334604685, 7.28762899546426866318493413724, 7.914242853678537265424321248442, 8.181609956392185039043808536273
