| L(s) = 1 | − 1.32·3-s + 5·5-s − 13.2·7-s − 25.2·9-s − 15.3·11-s − 80.8·13-s − 6.62·15-s − 25.5·17-s − 151.·19-s + 17.5·21-s + 23·23-s + 25·25-s + 69.2·27-s − 271.·29-s + 111.·31-s + 20.3·33-s − 66.1·35-s − 355.·37-s + 107.·39-s + 196.·41-s + 248.·43-s − 126.·45-s + 247.·47-s − 167.·49-s + 33.9·51-s + 238.·53-s − 76.8·55-s + ⋯ |
| L(s) = 1 | − 0.254·3-s + 0.447·5-s − 0.714·7-s − 0.934·9-s − 0.421·11-s − 1.72·13-s − 0.114·15-s − 0.365·17-s − 1.82·19-s + 0.182·21-s + 0.208·23-s + 0.200·25-s + 0.493·27-s − 1.73·29-s + 0.647·31-s + 0.107·33-s − 0.319·35-s − 1.57·37-s + 0.439·39-s + 0.749·41-s + 0.879·43-s − 0.418·45-s + 0.769·47-s − 0.489·49-s + 0.0931·51-s + 0.617·53-s − 0.188·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4630766140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4630766140\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 1.32T + 27T^{2} \) |
| 7 | \( 1 + 13.2T + 343T^{2} \) |
| 11 | \( 1 + 15.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 80.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 25.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 151.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 271.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 111.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 355.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 196.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 248.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 247.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 238.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 39.8T + 2.05e5T^{2} \) |
| 61 | \( 1 - 643.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 349.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 355.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 691.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 480.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 585.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.08e3T + 9.12e5T^{2} \) |
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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
−9.026061487708408989022976887126, −8.151922234126625870250121660370, −7.18060873499488964694166380560, −6.48846437875665065587305499270, −5.65585666666096899706898528440, −4.98800565452833424998474034837, −3.91961493031686679399789513793, −2.67914357935652659607440031123, −2.15700256942121628592465955769, −0.29589168698378361891305285881,
0.29589168698378361891305285881, 2.15700256942121628592465955769, 2.67914357935652659607440031123, 3.91961493031686679399789513793, 4.98800565452833424998474034837, 5.65585666666096899706898528440, 6.48846437875665065587305499270, 7.18060873499488964694166380560, 8.151922234126625870250121660370, 9.026061487708408989022976887126
