| L(s) = 1 | + 2-s + 2.61·3-s + 4-s + 2.61·6-s − 2.85·7-s + 8-s + 3.85·9-s − 5.23·11-s + 2.61·12-s + 1.23·13-s − 2.85·14-s + 16-s + 0.763·17-s + 3.85·18-s − 7.23·19-s − 7.47·21-s − 5.23·22-s + 7.61·23-s + 2.61·24-s + 1.23·26-s + 2.23·27-s − 2.85·28-s − 1.38·29-s + 3.70·31-s + 32-s − 13.7·33-s + 0.763·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.51·3-s + 0.5·4-s + 1.06·6-s − 1.07·7-s + 0.353·8-s + 1.28·9-s − 1.57·11-s + 0.755·12-s + 0.342·13-s − 0.762·14-s + 0.250·16-s + 0.185·17-s + 0.908·18-s − 1.66·19-s − 1.63·21-s − 1.11·22-s + 1.58·23-s + 0.534·24-s + 0.242·26-s + 0.430·27-s − 0.539·28-s − 0.256·29-s + 0.666·31-s + 0.176·32-s − 2.38·33-s + 0.131·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.483261604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.483261604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.61T + 3T^{2} \) |
| 7 | \( 1 + 2.85T + 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 - 7.61T + 23T^{2} \) |
| 29 | \( 1 + 1.38T + 29T^{2} \) |
| 31 | \( 1 - 3.70T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 3.38T + 41T^{2} \) |
| 43 | \( 1 + 0.145T + 43T^{2} \) |
| 47 | \( 1 - 8.32T + 47T^{2} \) |
| 53 | \( 1 + 4.94T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 + 6.47T + 67T^{2} \) |
| 71 | \( 1 + 0.763T + 71T^{2} \) |
| 73 | \( 1 - 1.23T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 4.09T + 83T^{2} \) |
| 89 | \( 1 - 8.09T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−12.75918560268028582349045230954, −10.99067531711996835128177591950, −10.12037889143558473235367876097, −9.064305940989654370347091639284, −8.139951127975277836300035370957, −7.17443803498718978915483211013, −5.97134695072252557723238839519, −4.45701006290169410259843594134, −3.18395430592504743280364659465, −2.49022908884304868479733842156,
2.49022908884304868479733842156, 3.18395430592504743280364659465, 4.45701006290169410259843594134, 5.97134695072252557723238839519, 7.17443803498718978915483211013, 8.139951127975277836300035370957, 9.064305940989654370347091639284, 10.12037889143558473235367876097, 10.99067531711996835128177591950, 12.75918560268028582349045230954
