| L(s) = 1 | + 2.38·2-s + 3·3-s − 2.29·4-s + 9.60·5-s + 7.16·6-s − 24.5·8-s + 9·9-s + 22.9·10-s − 11·11-s − 6.88·12-s − 38.2·13-s + 28.8·15-s − 40.3·16-s + 114.·17-s + 21.4·18-s + 57.0·19-s − 22.0·20-s − 26.2·22-s − 40.1·23-s − 73.7·24-s − 32.8·25-s − 91.4·26-s + 27·27-s − 71.8·29-s + 68.8·30-s + 220.·31-s + 100.·32-s + ⋯ |
| L(s) = 1 | + 0.844·2-s + 0.577·3-s − 0.286·4-s + 0.858·5-s + 0.487·6-s − 1.08·8-s + 0.333·9-s + 0.725·10-s − 0.301·11-s − 0.165·12-s − 0.817·13-s + 0.495·15-s − 0.630·16-s + 1.62·17-s + 0.281·18-s + 0.689·19-s − 0.246·20-s − 0.254·22-s − 0.364·23-s − 0.627·24-s − 0.262·25-s − 0.690·26-s + 0.192·27-s − 0.460·29-s + 0.418·30-s + 1.27·31-s + 0.553·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.211195200\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.211195200\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 2.38T + 8T^{2} \) |
| 5 | \( 1 - 9.60T + 125T^{2} \) |
| 13 | \( 1 + 38.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 114.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 57.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 40.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 71.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 220.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 99.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 452.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 6.92T + 7.95e4T^{2} \) |
| 47 | \( 1 + 74.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 35.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 115.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 3.61T + 2.26e5T^{2} \) |
| 67 | \( 1 - 453.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 129.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 315.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 491.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 851.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.52e3T + 9.12e5T^{2} \) |
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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
−9.297010464287557993224524762632, −8.126753722717244743304897337085, −7.55566344862016551022959762370, −6.33239163751441144613493262843, −5.58944167084198832239657435204, −4.97498308476976271477934613910, −3.95888189373193568116959468541, −3.05425344555600212952949864479, −2.27105080853187722274904343760, −0.856846748706542317162636384388,
0.856846748706542317162636384388, 2.27105080853187722274904343760, 3.05425344555600212952949864479, 3.95888189373193568116959468541, 4.97498308476976271477934613910, 5.58944167084198832239657435204, 6.33239163751441144613493262843, 7.55566344862016551022959762370, 8.126753722717244743304897337085, 9.297010464287557993224524762632
