| L(s) = 1 | − 2-s − 2.61·3-s + 4-s + 0.199·5-s + 2.61·6-s + 0.481·7-s − 8-s + 3.85·9-s − 0.199·10-s − 2.90·11-s − 2.61·12-s − 1.04·13-s − 0.481·14-s − 0.523·15-s + 16-s − 2.58·17-s − 3.85·18-s − 19-s + 0.199·20-s − 1.25·21-s + 2.90·22-s − 5.96·23-s + 2.61·24-s − 4.96·25-s + 1.04·26-s − 2.22·27-s + 0.481·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.51·3-s + 0.5·4-s + 0.0894·5-s + 1.06·6-s + 0.181·7-s − 0.353·8-s + 1.28·9-s − 0.0632·10-s − 0.876·11-s − 0.755·12-s − 0.289·13-s − 0.128·14-s − 0.135·15-s + 0.250·16-s − 0.627·17-s − 0.907·18-s − 0.229·19-s + 0.0447·20-s − 0.274·21-s + 0.619·22-s − 1.24·23-s + 0.534·24-s − 0.992·25-s + 0.205·26-s − 0.428·27-s + 0.0909·28-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
| \(L(1)\) |
\(\approx\) |
\(0.08334846984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08334846984\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 - 0.199T + 5T^{2} \) |
| 7 | \( 1 - 0.481T + 7T^{2} \) |
| 11 | \( 1 + 2.90T + 11T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 23 | \( 1 + 5.96T + 23T^{2} \) |
| 29 | \( 1 + 7.95T + 29T^{2} \) |
| 31 | \( 1 + 6.94T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 - 2.31T + 41T^{2} \) |
| 43 | \( 1 + 1.60T + 43T^{2} \) |
| 47 | \( 1 + 2.55T + 47T^{2} \) |
| 53 | \( 1 + 9.03T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 2.84T + 61T^{2} \) |
| 67 | \( 1 - 2.09T + 67T^{2} \) |
| 71 | \( 1 + 3.34T + 71T^{2} \) |
| 73 | \( 1 - 7.77T + 73T^{2} \) |
| 79 | \( 1 + 8.22T + 79T^{2} \) |
| 83 | \( 1 + 2.04T + 83T^{2} \) |
| 89 | \( 1 + 0.572T + 89T^{2} \) |
| 97 | \( 1 + 6.13T + 97T^{2} \) |
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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
−7.78722410681294493122588687391, −7.10726597557044413166017707886, −6.45961985635026297020934860558, −5.68112626257548827887985664785, −5.37585441376025975523438349379, −4.46360802685546757328513816954, −3.59858684881730500208941820154, −2.27850268613331284375212737034, −1.62861243951676975295963993593, −0.16768469904596962193468760997,
0.16768469904596962193468760997, 1.62861243951676975295963993593, 2.27850268613331284375212737034, 3.59858684881730500208941820154, 4.46360802685546757328513816954, 5.37585441376025975523438349379, 5.68112626257548827887985664785, 6.45961985635026297020934860558, 7.10726597557044413166017707886, 7.78722410681294493122588687391
