| L(s) = 1 | + 2-s − 0.0362·3-s + 4-s − 4.03·5-s − 0.0362·6-s − 1.31·7-s + 8-s − 2.99·9-s − 4.03·10-s − 2.27·11-s − 0.0362·12-s − 6.73·13-s − 1.31·14-s + 0.146·15-s + 16-s − 3.72·17-s − 2.99·18-s + 19-s − 4.03·20-s + 0.0477·21-s − 2.27·22-s + 1.64·23-s − 0.0362·24-s + 11.3·25-s − 6.73·26-s + 0.217·27-s − 1.31·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0209·3-s + 0.5·4-s − 1.80·5-s − 0.0148·6-s − 0.497·7-s + 0.353·8-s − 0.999·9-s − 1.27·10-s − 0.685·11-s − 0.0104·12-s − 1.86·13-s − 0.351·14-s + 0.0378·15-s + 0.250·16-s − 0.903·17-s − 0.706·18-s + 0.229·19-s − 0.902·20-s + 0.0104·21-s − 0.485·22-s + 0.342·23-s − 0.00740·24-s + 2.26·25-s − 1.32·26-s + 0.0418·27-s − 0.248·28-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
| \(L(1)\) |
\(\approx\) |
\(0.09855671541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09855671541\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.0362T + 3T^{2} \) |
| 5 | \( 1 + 4.03T + 5T^{2} \) |
| 7 | \( 1 + 1.31T + 7T^{2} \) |
| 11 | \( 1 + 2.27T + 11T^{2} \) |
| 13 | \( 1 + 6.73T + 13T^{2} \) |
| 17 | \( 1 + 3.72T + 17T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 + 5.19T + 29T^{2} \) |
| 31 | \( 1 + 8.46T + 31T^{2} \) |
| 37 | \( 1 + 7.42T + 37T^{2} \) |
| 41 | \( 1 - 9.87T + 41T^{2} \) |
| 43 | \( 1 + 7.38T + 43T^{2} \) |
| 47 | \( 1 + 9.42T + 47T^{2} \) |
| 53 | \( 1 + 5.92T + 53T^{2} \) |
| 59 | \( 1 + 3.76T + 59T^{2} \) |
| 61 | \( 1 - 3.65T + 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 - 0.945T + 71T^{2} \) |
| 73 | \( 1 - 2.24T + 73T^{2} \) |
| 79 | \( 1 + 4.29T + 79T^{2} \) |
| 83 | \( 1 - 1.30T + 83T^{2} \) |
| 89 | \( 1 + 8.34T + 89T^{2} \) |
| 97 | \( 1 + 4.60T + 97T^{2} \) |
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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
−7.74648745585954113544244611270, −7.11688020124220114127322103650, −6.65001064969641897366639539652, −5.43936112774462000459318235274, −5.02932754674816653581786345740, −4.30205987314155921312606887661, −3.43691600178381903928228026664, −2.99084641185051190302272380856, −2.11418116932039801416990828355, −0.12671454312475361253380727024,
0.12671454312475361253380727024, 2.11418116932039801416990828355, 2.99084641185051190302272380856, 3.43691600178381903928228026664, 4.30205987314155921312606887661, 5.02932754674816653581786345740, 5.43936112774462000459318235274, 6.65001064969641897366639539652, 7.11688020124220114127322103650, 7.74648745585954113544244611270
