| L(s) = 1 | + 2-s − 2.40·3-s + 4-s − 2.40·6-s + 0.706·7-s + 8-s + 2.79·9-s + 0.747·11-s − 2.40·12-s + 1.29·13-s + 0.706·14-s + 16-s − 5.50·17-s + 2.79·18-s + 2.44·19-s − 1.70·21-s + 0.747·22-s + 23-s − 2.40·24-s + 1.29·26-s + 0.483·27-s + 0.706·28-s + 5.72·29-s + 7.52·31-s + 32-s − 1.79·33-s − 5.50·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.39·3-s + 0.5·4-s − 0.983·6-s + 0.267·7-s + 0.353·8-s + 0.933·9-s + 0.225·11-s − 0.695·12-s + 0.358·13-s + 0.188·14-s + 0.250·16-s − 1.33·17-s + 0.659·18-s + 0.561·19-s − 0.371·21-s + 0.159·22-s + 0.208·23-s − 0.491·24-s + 0.253·26-s + 0.0930·27-s + 0.133·28-s + 1.06·29-s + 1.35·31-s + 0.176·32-s − 0.313·33-s − 0.943·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.632774205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632774205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2.40T + 3T^{2} \) |
| 7 | \( 1 - 0.706T + 7T^{2} \) |
| 11 | \( 1 - 0.747T + 11T^{2} \) |
| 13 | \( 1 - 1.29T + 13T^{2} \) |
| 17 | \( 1 + 5.50T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 29 | \( 1 - 5.72T + 29T^{2} \) |
| 31 | \( 1 - 7.52T + 31T^{2} \) |
| 37 | \( 1 + 5.07T + 37T^{2} \) |
| 41 | \( 1 - 10.0T + 41T^{2} \) |
| 43 | \( 1 + 5.34T + 43T^{2} \) |
| 47 | \( 1 - 7.90T + 47T^{2} \) |
| 53 | \( 1 - 5.84T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 - 1.57T + 61T^{2} \) |
| 67 | \( 1 - 5.25T + 67T^{2} \) |
| 71 | \( 1 - 2.68T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 6.55T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 - 1.50T + 97T^{2} \) |
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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
−10.18480582052081388935158594291, −9.006820898077904391658123167813, −8.020055110685366635570764988168, −6.77199311933913387101245185046, −6.46146567477387419646511638450, −5.44334870509393962082054802980, −4.80399577654227329769222663421, −3.94421826413573150555373481893, −2.50742918512340259068226136220, −0.962055933637588161213534009753,
0.962055933637588161213534009753, 2.50742918512340259068226136220, 3.94421826413573150555373481893, 4.80399577654227329769222663421, 5.44334870509393962082054802980, 6.46146567477387419646511638450, 6.77199311933913387101245185046, 8.020055110685366635570764988168, 9.006820898077904391658123167813, 10.18480582052081388935158594291
