| L(s) = 1 | − 2·2-s + 4·4-s + 20·5-s + 2·7-s − 8·8-s − 40·10-s + 52·11-s + 43·13-s − 4·14-s + 16·16-s + 50·17-s − 74·19-s + 80·20-s − 104·22-s + 23·23-s + 275·25-s − 86·26-s + 8·28-s + 7·29-s − 273·31-s − 32·32-s − 100·34-s + 40·35-s − 4·37-s + 148·38-s − 160·40-s − 123·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.107·7-s − 0.353·8-s − 1.26·10-s + 1.42·11-s + 0.917·13-s − 0.0763·14-s + 1/4·16-s + 0.713·17-s − 0.893·19-s + 0.894·20-s − 1.00·22-s + 0.208·23-s + 11/5·25-s − 0.648·26-s + 0.0539·28-s + 0.0448·29-s − 1.58·31-s − 0.176·32-s − 0.504·34-s + 0.193·35-s − 0.0177·37-s + 0.631·38-s − 0.632·40-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.234673921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.234673921\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 23 | \( 1 - p T \) |
| good | 5 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 52 T + p^{3} T^{2} \) |
| 13 | \( 1 - 43 T + p^{3} T^{2} \) |
| 17 | \( 1 - 50 T + p^{3} T^{2} \) |
| 19 | \( 1 + 74 T + p^{3} T^{2} \) |
| 29 | \( 1 - 7 T + p^{3} T^{2} \) |
| 31 | \( 1 + 273 T + p^{3} T^{2} \) |
| 37 | \( 1 + 4 T + p^{3} T^{2} \) |
| 41 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 152 T + p^{3} T^{2} \) |
| 47 | \( 1 + 75 T + p^{3} T^{2} \) |
| 53 | \( 1 + 86 T + p^{3} T^{2} \) |
| 59 | \( 1 - 444 T + p^{3} T^{2} \) |
| 61 | \( 1 - 262 T + p^{3} T^{2} \) |
| 67 | \( 1 - 764 T + p^{3} T^{2} \) |
| 71 | \( 1 - 21 T + p^{3} T^{2} \) |
| 73 | \( 1 - 681 T + p^{3} T^{2} \) |
| 79 | \( 1 - 426 T + p^{3} T^{2} \) |
| 83 | \( 1 + 902 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1272 T + p^{3} T^{2} \) |
| 97 | \( 1 + 342 T + p^{3} T^{2} \) |
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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
−10.61534446896138448299377143239, −9.730944728547469007201049129239, −9.146063389954310601653318066666, −8.357788841385969104621887130030, −6.81873190724901800605663122799, −6.25369824678468603078062912538, −5.30741060068846072662647073995, −3.60334927597568441646932613616, −2.03204239413217092562673262477, −1.22269736217940629711142619776,
1.22269736217940629711142619776, 2.03204239413217092562673262477, 3.60334927597568441646932613616, 5.30741060068846072662647073995, 6.25369824678468603078062912538, 6.81873190724901800605663122799, 8.357788841385969104621887130030, 9.146063389954310601653318066666, 9.730944728547469007201049129239, 10.61534446896138448299377143239
