| L(s) = 1 | + 1.18·2-s − 0.597·4-s − 2.95·5-s + 3.03·7-s − 3.07·8-s − 3.49·10-s − 11-s − 2.06·13-s + 3.59·14-s − 2.44·16-s + 3.46·17-s + 2.26·19-s + 1.76·20-s − 1.18·22-s − 5.95·23-s + 3.72·25-s − 2.44·26-s − 1.81·28-s + 6.05·29-s − 3.70·31-s + 3.25·32-s + 4.10·34-s − 8.97·35-s − 9.05·37-s + 2.68·38-s + 9.08·40-s − 4.30·41-s + ⋯ |
| L(s) = 1 | + 0.837·2-s − 0.298·4-s − 1.32·5-s + 1.14·7-s − 1.08·8-s − 1.10·10-s − 0.301·11-s − 0.572·13-s + 0.961·14-s − 0.612·16-s + 0.840·17-s + 0.520·19-s + 0.394·20-s − 0.252·22-s − 1.24·23-s + 0.744·25-s − 0.479·26-s − 0.343·28-s + 1.12·29-s − 0.666·31-s + 0.575·32-s + 0.703·34-s − 1.51·35-s − 1.48·37-s + 0.435·38-s + 1.43·40-s − 0.672·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8019 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.590084786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.590084786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.18T + 2T^{2} \) |
| 5 | \( 1 + 2.95T + 5T^{2} \) |
| 7 | \( 1 - 3.03T + 7T^{2} \) |
| 13 | \( 1 + 2.06T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 2.26T + 19T^{2} \) |
| 23 | \( 1 + 5.95T + 23T^{2} \) |
| 29 | \( 1 - 6.05T + 29T^{2} \) |
| 31 | \( 1 + 3.70T + 31T^{2} \) |
| 37 | \( 1 + 9.05T + 37T^{2} \) |
| 41 | \( 1 + 4.30T + 41T^{2} \) |
| 43 | \( 1 - 3.78T + 43T^{2} \) |
| 47 | \( 1 + 2.49T + 47T^{2} \) |
| 53 | \( 1 - 8.53T + 53T^{2} \) |
| 59 | \( 1 + 5.83T + 59T^{2} \) |
| 61 | \( 1 + 7.77T + 61T^{2} \) |
| 67 | \( 1 - 4.81T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 1.76T + 79T^{2} \) |
| 83 | \( 1 - 8.91T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
| 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
−7.72934493489123518456115003456, −7.38694646014962073077372526998, −6.30822252836216742683106310821, −5.42094836689656400570177403044, −4.93801635012251863003608597270, −4.35364345463097519457062351095, −3.65334568624134608226175228687, −3.04661340841888579834551263235, −1.87231882355729897165785527726, −0.54364934889154615080175399459,
0.54364934889154615080175399459, 1.87231882355729897165785527726, 3.04661340841888579834551263235, 3.65334568624134608226175228687, 4.35364345463097519457062351095, 4.93801635012251863003608597270, 5.42094836689656400570177403044, 6.30822252836216742683106310821, 7.38694646014962073077372526998, 7.72934493489123518456115003456
