| L(s) = 1 | + 0.514·2-s − 1.73·4-s − 5-s − 4.44·7-s − 1.92·8-s − 0.514·10-s − 5.37·11-s + 13-s − 2.29·14-s + 2.47·16-s + 4.15·17-s − 2.50·19-s + 1.73·20-s − 2.76·22-s + 2.60·23-s + 25-s + 0.514·26-s + 7.71·28-s − 3.18·29-s + 4.19·31-s + 5.12·32-s + 2.14·34-s + 4.44·35-s − 5.57·37-s − 1.28·38-s + 1.92·40-s − 4.43·41-s + ⋯ |
| L(s) = 1 | + 0.364·2-s − 0.867·4-s − 0.447·5-s − 1.68·7-s − 0.679·8-s − 0.162·10-s − 1.61·11-s + 0.277·13-s − 0.612·14-s + 0.619·16-s + 1.00·17-s − 0.574·19-s + 0.387·20-s − 0.589·22-s + 0.543·23-s + 0.200·25-s + 0.100·26-s + 1.45·28-s − 0.591·29-s + 0.753·31-s + 0.905·32-s + 0.367·34-s + 0.752·35-s − 0.916·37-s − 0.209·38-s + 0.304·40-s − 0.692·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6926255305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6926255305\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.514T + 2T^{2} \) |
| 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 + 5.37T + 11T^{2} \) |
| 17 | \( 1 - 4.15T + 17T^{2} \) |
| 19 | \( 1 + 2.50T + 19T^{2} \) |
| 23 | \( 1 - 2.60T + 23T^{2} \) |
| 29 | \( 1 + 3.18T + 29T^{2} \) |
| 31 | \( 1 - 4.19T + 31T^{2} \) |
| 37 | \( 1 + 5.57T + 37T^{2} \) |
| 41 | \( 1 + 4.43T + 41T^{2} \) |
| 43 | \( 1 - 8.18T + 43T^{2} \) |
| 47 | \( 1 - 1.07T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 6.66T + 61T^{2} \) |
| 67 | \( 1 + 3.83T + 67T^{2} \) |
| 71 | \( 1 - 5.54T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 + 6.82T + 79T^{2} \) |
| 83 | \( 1 + 7.79T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 + 1.49T + 97T^{2} \) |
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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
−9.261277649640696221286098904081, −8.597625808339433792588657457921, −7.75030530787488759967781791561, −6.90245153032170970821123589230, −5.84901861345217937791045643341, −5.32955305877675515778488355306, −4.21742194633125054882821443161, −3.38069433991352329174286335781, −2.74046381838120927176437976394, −0.51600412710805549937142657545,
0.51600412710805549937142657545, 2.74046381838120927176437976394, 3.38069433991352329174286335781, 4.21742194633125054882821443161, 5.32955305877675515778488355306, 5.84901861345217937791045643341, 6.90245153032170970821123589230, 7.75030530787488759967781791561, 8.597625808339433792588657457921, 9.261277649640696221286098904081
