| L(s) = 1 | − 2-s − 2.46·3-s + 4-s − 0.0638·5-s + 2.46·6-s + 3.24·7-s − 8-s + 3.06·9-s + 0.0638·10-s + 1.81·11-s − 2.46·12-s − 6.18·13-s − 3.24·14-s + 0.157·15-s + 16-s + 2.99·17-s − 3.06·18-s − 19-s − 0.0638·20-s − 7.98·21-s − 1.81·22-s − 0.142·23-s + 2.46·24-s − 4.99·25-s + 6.18·26-s − 0.171·27-s + 3.24·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.42·3-s + 0.5·4-s − 0.0285·5-s + 1.00·6-s + 1.22·7-s − 0.353·8-s + 1.02·9-s + 0.0201·10-s + 0.547·11-s − 0.711·12-s − 1.71·13-s − 0.866·14-s + 0.0405·15-s + 0.250·16-s + 0.726·17-s − 0.723·18-s − 0.229·19-s − 0.0142·20-s − 1.74·21-s − 0.386·22-s − 0.0296·23-s + 0.502·24-s − 0.999·25-s + 1.21·26-s − 0.0329·27-s + 0.612·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 2.46T + 3T^{2} \) |
| 5 | \( 1 + 0.0638T + 5T^{2} \) |
| 7 | \( 1 - 3.24T + 7T^{2} \) |
| 11 | \( 1 - 1.81T + 11T^{2} \) |
| 13 | \( 1 + 6.18T + 13T^{2} \) |
| 17 | \( 1 - 2.99T + 17T^{2} \) |
| 23 | \( 1 + 0.142T + 23T^{2} \) |
| 29 | \( 1 - 1.88T + 29T^{2} \) |
| 31 | \( 1 - 8.38T + 31T^{2} \) |
| 37 | \( 1 + 0.447T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 9.43T + 43T^{2} \) |
| 47 | \( 1 + 4.96T + 47T^{2} \) |
| 53 | \( 1 - 3.48T + 53T^{2} \) |
| 59 | \( 1 - 2.98T + 59T^{2} \) |
| 61 | \( 1 - 4.57T + 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 11.2T + 73T^{2} \) |
| 79 | \( 1 + 3.92T + 79T^{2} \) |
| 83 | \( 1 - 4.00T + 83T^{2} \) |
| 89 | \( 1 + 8.86T + 89T^{2} \) |
| 97 | \( 1 - 2.20T + 97T^{2} \) |
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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
−7.45457405495702055928523388069, −6.88057162489741451494404331545, −6.09557138375964308131261467136, −5.44945219836996657155304837222, −4.80080454824369754978436680033, −4.25690955654103478347112493599, −2.87822884703383775170015149825, −1.87930957825620555274784426081, −1.05335987817910327696220501808, 0,
1.05335987817910327696220501808, 1.87930957825620555274784426081, 2.87822884703383775170015149825, 4.25690955654103478347112493599, 4.80080454824369754978436680033, 5.44945219836996657155304837222, 6.09557138375964308131261467136, 6.88057162489741451494404331545, 7.45457405495702055928523388069
