| L(s) = 1 | − 2.55·2-s + 3-s + 4.53·4-s + 0.170·5-s − 2.55·6-s − 7-s − 6.48·8-s + 9-s − 0.435·10-s + 0.356·11-s + 4.53·12-s − 1.97·13-s + 2.55·14-s + 0.170·15-s + 7.49·16-s + 2.43·17-s − 2.55·18-s − 7.54·19-s + 0.772·20-s − 21-s − 0.911·22-s − 4.32·23-s − 6.48·24-s − 4.97·25-s + 5.03·26-s + 27-s − 4.53·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 0.577·3-s + 2.26·4-s + 0.0762·5-s − 1.04·6-s − 0.377·7-s − 2.29·8-s + 0.333·9-s − 0.137·10-s + 0.107·11-s + 1.30·12-s − 0.546·13-s + 0.683·14-s + 0.0440·15-s + 1.87·16-s + 0.590·17-s − 0.602·18-s − 1.73·19-s + 0.172·20-s − 0.218·21-s − 0.194·22-s − 0.901·23-s − 1.32·24-s − 0.994·25-s + 0.988·26-s + 0.192·27-s − 0.857·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7462153530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7462153530\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 + T \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 5 | \( 1 - 0.170T + 5T^{2} \) |
| 11 | \( 1 - 0.356T + 11T^{2} \) |
| 13 | \( 1 + 1.97T + 13T^{2} \) |
| 17 | \( 1 - 2.43T + 17T^{2} \) |
| 19 | \( 1 + 7.54T + 19T^{2} \) |
| 23 | \( 1 + 4.32T + 23T^{2} \) |
| 29 | \( 1 - 2.57T + 29T^{2} \) |
| 31 | \( 1 - 2.19T + 31T^{2} \) |
| 37 | \( 1 - 4.37T + 37T^{2} \) |
| 41 | \( 1 - 7.20T + 41T^{2} \) |
| 43 | \( 1 + 4.73T + 43T^{2} \) |
| 47 | \( 1 + 5.18T + 47T^{2} \) |
| 53 | \( 1 - 10.6T + 53T^{2} \) |
| 59 | \( 1 + 5.46T + 59T^{2} \) |
| 61 | \( 1 + 3.72T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 3.92T + 71T^{2} \) |
| 73 | \( 1 + 3.78T + 73T^{2} \) |
| 79 | \( 1 + 0.543T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 18.0T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.436377056757649272497009859456, −7.951024677231677840956041683914, −7.33810654190362572803086678309, −6.46329255425190846340272123233, −5.99530423612638789053771760887, −4.55632293693926424774957284872, −3.53432294594041944277940543822, −2.44031358498321426099006431452, −1.93111272187844367183736521336, −0.60893880344609619764008762645,
0.60893880344609619764008762645, 1.93111272187844367183736521336, 2.44031358498321426099006431452, 3.53432294594041944277940543822, 4.55632293693926424774957284872, 5.99530423612638789053771760887, 6.46329255425190846340272123233, 7.33810654190362572803086678309, 7.951024677231677840956041683914, 8.436377056757649272497009859456
