| L(s) = 1 | + 10·3-s + 8·5-s − 7·7-s + 73·9-s + 40·11-s + 12·13-s + 80·15-s − 58·17-s − 26·19-s − 70·21-s − 64·23-s − 61·25-s + 460·27-s + 62·29-s + 252·31-s + 400·33-s − 56·35-s − 26·37-s + 120·39-s + 6·41-s − 416·43-s + 584·45-s − 396·47-s + 49·49-s − 580·51-s + 450·53-s + 320·55-s + ⋯ |
| L(s) = 1 | + 1.92·3-s + 0.715·5-s − 0.377·7-s + 2.70·9-s + 1.09·11-s + 0.256·13-s + 1.37·15-s − 0.827·17-s − 0.313·19-s − 0.727·21-s − 0.580·23-s − 0.487·25-s + 3.27·27-s + 0.397·29-s + 1.46·31-s + 2.11·33-s − 0.270·35-s − 0.115·37-s + 0.492·39-s + 0.0228·41-s − 1.47·43-s + 1.93·45-s − 1.22·47-s + 1/7·49-s − 1.59·51-s + 1.16·53-s + 0.784·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.505478247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.505478247\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 - 10 T + p^{3} T^{2} \) |
| 5 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 40 T + p^{3} T^{2} \) |
| 13 | \( 1 - 12 T + p^{3} T^{2} \) |
| 17 | \( 1 + 58 T + p^{3} T^{2} \) |
| 19 | \( 1 + 26 T + p^{3} T^{2} \) |
| 23 | \( 1 + 64 T + p^{3} T^{2} \) |
| 29 | \( 1 - 62 T + p^{3} T^{2} \) |
| 31 | \( 1 - 252 T + p^{3} T^{2} \) |
| 37 | \( 1 + 26 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 416 T + p^{3} T^{2} \) |
| 47 | \( 1 + 396 T + p^{3} T^{2} \) |
| 53 | \( 1 - 450 T + p^{3} T^{2} \) |
| 59 | \( 1 + 274 T + p^{3} T^{2} \) |
| 61 | \( 1 - 576 T + p^{3} T^{2} \) |
| 67 | \( 1 - 476 T + p^{3} T^{2} \) |
| 71 | \( 1 + 448 T + p^{3} T^{2} \) |
| 73 | \( 1 + 158 T + p^{3} T^{2} \) |
| 79 | \( 1 + 936 T + p^{3} T^{2} \) |
| 83 | \( 1 + 530 T + p^{3} T^{2} \) |
| 89 | \( 1 + 390 T + p^{3} T^{2} \) |
| 97 | \( 1 - 214 T + p^{3} T^{2} \) |
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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
−10.09353233571804528834858042313, −9.755787149080613507792718166859, −8.770849430803615340030768324084, −8.309640590597552077329797419834, −7.01888141706865606453230044766, −6.29373133754404193390326139054, −4.47809848513677776869447353399, −3.58223832596677951757011782842, −2.47363566437369208362842139504, −1.51145227866669935189979176355,
1.51145227866669935189979176355, 2.47363566437369208362842139504, 3.58223832596677951757011782842, 4.47809848513677776869447353399, 6.29373133754404193390326139054, 7.01888141706865606453230044766, 8.309640590597552077329797419834, 8.770849430803615340030768324084, 9.755787149080613507792718166859, 10.09353233571804528834858042313
