| L(s) = 1 | + 2.27·3-s − 2.08·5-s + 1.11·7-s + 2.16·9-s + 4.08·11-s + 0.839·13-s − 4.74·15-s + 3.11·17-s + 2.97·19-s + 2.52·21-s − 23-s − 0.635·25-s − 1.90·27-s + 9.01·29-s − 0.315·31-s + 9.28·33-s − 2.32·35-s − 2.08·37-s + 1.90·39-s + 11.3·41-s + 0.478·43-s − 4.51·45-s − 8.22·47-s − 5.76·49-s + 7.06·51-s + 0.434·53-s − 8.54·55-s + ⋯ |
| L(s) = 1 | + 1.31·3-s − 0.934·5-s + 0.419·7-s + 0.720·9-s + 1.23·11-s + 0.232·13-s − 1.22·15-s + 0.754·17-s + 0.683·19-s + 0.550·21-s − 0.208·23-s − 0.127·25-s − 0.366·27-s + 1.67·29-s − 0.0566·31-s + 1.61·33-s − 0.392·35-s − 0.343·37-s + 0.305·39-s + 1.77·41-s + 0.0729·43-s − 0.672·45-s − 1.20·47-s − 0.823·49-s + 0.989·51-s + 0.0597·53-s − 1.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.260388833\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.260388833\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2.27T + 3T^{2} \) |
| 5 | \( 1 + 2.08T + 5T^{2} \) |
| 7 | \( 1 - 1.11T + 7T^{2} \) |
| 11 | \( 1 - 4.08T + 11T^{2} \) |
| 13 | \( 1 - 0.839T + 13T^{2} \) |
| 17 | \( 1 - 3.11T + 17T^{2} \) |
| 19 | \( 1 - 2.97T + 19T^{2} \) |
| 29 | \( 1 - 9.01T + 29T^{2} \) |
| 31 | \( 1 + 0.315T + 31T^{2} \) |
| 37 | \( 1 + 2.08T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 0.478T + 43T^{2} \) |
| 47 | \( 1 + 8.22T + 47T^{2} \) |
| 53 | \( 1 - 0.434T + 53T^{2} \) |
| 59 | \( 1 + 2.86T + 59T^{2} \) |
| 61 | \( 1 - 3.56T + 61T^{2} \) |
| 67 | \( 1 + 6.31T + 67T^{2} \) |
| 71 | \( 1 + 8.12T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 0.454T + 83T^{2} \) |
| 89 | \( 1 - 4.32T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−10.18380591414681916635468475479, −9.322797339888418371684847733330, −8.572728818683065819566748263897, −7.924714031539183771241817647483, −7.24930048748986167672489812211, −6.04633584906049383844911402322, −4.56694884387657725901872265801, −3.73509076012436670950076095902, −2.92430089553604065968945502183, −1.38748541920227814240164103699,
1.38748541920227814240164103699, 2.92430089553604065968945502183, 3.73509076012436670950076095902, 4.56694884387657725901872265801, 6.04633584906049383844911402322, 7.24930048748986167672489812211, 7.924714031539183771241817647483, 8.572728818683065819566748263897, 9.322797339888418371684847733330, 10.18380591414681916635468475479
