| L(s) = 1 | + 2-s + 4-s + 3·5-s + 7-s + 8-s + 3·10-s + 6·11-s + 2·13-s + 14-s + 16-s − 6·17-s − 7·19-s + 3·20-s + 6·22-s − 3·23-s + 4·25-s + 2·26-s + 28-s − 6·29-s + 2·31-s + 32-s − 6·34-s + 3·35-s + 2·37-s − 7·38-s + 3·40-s + 2·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s + 0.353·8-s + 0.948·10-s + 1.80·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 1.60·19-s + 0.670·20-s + 1.27·22-s − 0.625·23-s + 4/5·25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s + 0.359·31-s + 0.176·32-s − 1.02·34-s + 0.507·35-s + 0.328·37-s − 1.13·38-s + 0.474·40-s + 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.432508916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.432508916\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.751640155951921812390830533275, −9.073822195109535382569774013470, −8.331847866731550649759204454275, −6.84669532198488939637919288494, −6.36293944731828035024321117773, −5.73900595462088482953892956618, −4.49364756499371052867692865672, −3.87247358034755873056669071066, −2.28395108895772497375168075701, −1.60097986545961424167062353389,
1.60097986545961424167062353389, 2.28395108895772497375168075701, 3.87247358034755873056669071066, 4.49364756499371052867692865672, 5.73900595462088482953892956618, 6.36293944731828035024321117773, 6.84669532198488939637919288494, 8.331847866731550649759204454275, 9.073822195109535382569774013470, 9.751640155951921812390830533275
