| L(s) = 1 | + 5.12·2-s − 3·3-s + 18.2·4-s − 15.3·6-s + 21.7·7-s + 52.5·8-s + 9·9-s + 57.1·11-s − 54.7·12-s + 41.2·13-s + 111.·14-s + 123.·16-s − 73.0·17-s + 46.1·18-s − 0.658·19-s − 65.2·21-s + 292.·22-s + 96.1·23-s − 157.·24-s + 211.·26-s − 27·27-s + 397.·28-s − 29·29-s − 2.01·31-s + 210.·32-s − 171.·33-s − 374.·34-s + ⋯ |
| L(s) = 1 | + 1.81·2-s − 0.577·3-s + 2.28·4-s − 1.04·6-s + 1.17·7-s + 2.32·8-s + 0.333·9-s + 1.56·11-s − 1.31·12-s + 0.879·13-s + 2.12·14-s + 1.92·16-s − 1.04·17-s + 0.603·18-s − 0.00795·19-s − 0.678·21-s + 2.83·22-s + 0.871·23-s − 1.34·24-s + 1.59·26-s − 0.192·27-s + 2.68·28-s − 0.185·29-s − 0.0116·31-s + 1.16·32-s − 0.904·33-s − 1.88·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.673669571\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.673669571\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 - 5.12T + 8T^{2} \) |
| 7 | \( 1 - 21.7T + 343T^{2} \) |
| 11 | \( 1 - 57.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 73.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 0.658T + 6.85e3T^{2} \) |
| 23 | \( 1 - 96.1T + 1.21e4T^{2} \) |
| 31 | \( 1 + 2.01T + 2.97e4T^{2} \) |
| 37 | \( 1 - 315.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 219.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 81.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 440.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 65.8T + 1.48e5T^{2} \) |
| 59 | \( 1 + 551.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 149.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 888.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 570.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 664.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 221.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 740.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 895.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 705.T + 9.12e5T^{2} \) |
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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
−8.606879881577556684759855850232, −7.58822449889234311744360440161, −6.65835190754515026953282859296, −6.27271243771101128197129776538, −5.41690264561742190521818907029, −4.51574785544112407133915138280, −4.22364159672046884991834684098, −3.19064601303826508231792832493, −1.91985974012151670899477482248, −1.17434942713417367717868308659,
1.17434942713417367717868308659, 1.91985974012151670899477482248, 3.19064601303826508231792832493, 4.22364159672046884991834684098, 4.51574785544112407133915138280, 5.41690264561742190521818907029, 6.27271243771101128197129776538, 6.65835190754515026953282859296, 7.58822449889234311744360440161, 8.606879881577556684759855850232
