| L(s) = 1 | − 4.31·2-s + 3·3-s + 10.5·4-s − 12.9·6-s − 9.06·7-s − 11.1·8-s + 9·9-s − 11·11-s + 31.7·12-s + 40.4·13-s + 39.0·14-s − 36.5·16-s + 76.4·17-s − 38.8·18-s − 44.7·19-s − 27.1·21-s + 47.4·22-s − 50.4·23-s − 33.5·24-s − 174.·26-s + 27·27-s − 96.0·28-s − 81.1·29-s + 100.·31-s + 246.·32-s − 33·33-s − 329.·34-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 0.577·3-s + 1.32·4-s − 0.880·6-s − 0.489·7-s − 0.494·8-s + 0.333·9-s − 0.301·11-s + 0.764·12-s + 0.862·13-s + 0.746·14-s − 0.570·16-s + 1.09·17-s − 0.508·18-s − 0.540·19-s − 0.282·21-s + 0.459·22-s − 0.457·23-s − 0.285·24-s − 1.31·26-s + 0.192·27-s − 0.648·28-s − 0.519·29-s + 0.580·31-s + 1.36·32-s − 0.174·33-s − 1.66·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.040580705\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040580705\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 4.31T + 8T^{2} \) |
| 7 | \( 1 + 9.06T + 343T^{2} \) |
| 13 | \( 1 - 40.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 76.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 44.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 50.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 81.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 100.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 38.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 18.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 40.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 36.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 652.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 744.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 516.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 430.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 272.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 692.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 595.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 706.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.31e3T + 9.12e5T^{2} \) |
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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
−9.805108285673020634076609715995, −8.926147954366615985650825020239, −8.301899296195177733082389015467, −7.61160520789008396008813320285, −6.73243051967502769838747794636, −5.72505618031312067517623302567, −4.17680316375307873724255855392, −3.03093645528454575805517474716, −1.85362445984197760772672484576, −0.70277753253125740251830412551,
0.70277753253125740251830412551, 1.85362445984197760772672484576, 3.03093645528454575805517474716, 4.17680316375307873724255855392, 5.72505618031312067517623302567, 6.73243051967502769838747794636, 7.61160520789008396008813320285, 8.301899296195177733082389015467, 8.926147954366615985650825020239, 9.805108285673020634076609715995
