| L(s) = 1 | − 2-s − 1.23·3-s + 4-s + 3.23·5-s + 1.23·6-s − 7-s − 8-s − 1.47·9-s − 3.23·10-s + 0.763·11-s − 1.23·12-s + 6.47·13-s + 14-s − 4.00·15-s + 16-s + 17-s + 1.47·18-s + 0.472·19-s + 3.23·20-s + 1.23·21-s − 0.763·22-s + 8·23-s + 1.23·24-s + 5.47·25-s − 6.47·26-s + 5.52·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.713·3-s + 0.5·4-s + 1.44·5-s + 0.504·6-s − 0.377·7-s − 0.353·8-s − 0.490·9-s − 1.02·10-s + 0.230·11-s − 0.356·12-s + 1.79·13-s + 0.267·14-s − 1.03·15-s + 0.250·16-s + 0.242·17-s + 0.346·18-s + 0.108·19-s + 0.723·20-s + 0.269·21-s − 0.162·22-s + 1.66·23-s + 0.252·24-s + 1.09·25-s − 1.26·26-s + 1.06·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9074036160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9074036160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 19 | \( 1 - 0.472T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 7.70T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 1.23T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 - 6.47T + 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 3.23T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−11.92612142471837471706435068419, −10.90494291763676779852377629713, −10.36006400201378464173297109921, −9.173706923637329481742301738429, −8.610264838777983408651887538747, −6.83921001446596737633967757560, −6.11347826531456823537239071134, −5.31932687459869695779278148644, −3.13391409231110049395176238944, −1.36349622199492200919804644331,
1.36349622199492200919804644331, 3.13391409231110049395176238944, 5.31932687459869695779278148644, 6.11347826531456823537239071134, 6.83921001446596737633967757560, 8.610264838777983408651887538747, 9.173706923637329481742301738429, 10.36006400201378464173297109921, 10.90494291763676779852377629713, 11.92612142471837471706435068419
