| L(s) = 1 | − 6·2-s + 9·3-s + 4·4-s − 78·5-s − 54·6-s + 168·8-s + 81·9-s + 468·10-s + 444·11-s + 36·12-s + 442·13-s − 702·15-s − 1.13e3·16-s + 126·17-s − 486·18-s − 2.68e3·19-s − 312·20-s − 2.66e3·22-s + 4.20e3·23-s + 1.51e3·24-s + 2.95e3·25-s − 2.65e3·26-s + 729·27-s − 5.44e3·29-s + 4.21e3·30-s − 80·31-s + 1.44e3·32-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.577·3-s + 1/8·4-s − 1.39·5-s − 0.612·6-s + 0.928·8-s + 1/3·9-s + 1.47·10-s + 1.10·11-s + 0.0721·12-s + 0.725·13-s − 0.805·15-s − 1.10·16-s + 0.105·17-s − 0.353·18-s − 1.70·19-s − 0.174·20-s − 1.17·22-s + 1.65·23-s + 0.535·24-s + 0.946·25-s − 0.769·26-s + 0.192·27-s − 1.20·29-s + 0.854·30-s − 0.0149·31-s + 0.248·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3 p T + p^{5} T^{2} \) |
| 5 | \( 1 + 78 T + p^{5} T^{2} \) |
| 11 | \( 1 - 444 T + p^{5} T^{2} \) |
| 13 | \( 1 - 34 p T + p^{5} T^{2} \) |
| 17 | \( 1 - 126 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2684 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4200 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5442 T + p^{5} T^{2} \) |
| 31 | \( 1 + 80 T + p^{5} T^{2} \) |
| 37 | \( 1 + 5434 T + p^{5} T^{2} \) |
| 41 | \( 1 + 7962 T + p^{5} T^{2} \) |
| 43 | \( 1 + 268 p T + p^{5} T^{2} \) |
| 47 | \( 1 - 13920 T + p^{5} T^{2} \) |
| 53 | \( 1 + 9594 T + p^{5} T^{2} \) |
| 59 | \( 1 + 27492 T + p^{5} T^{2} \) |
| 61 | \( 1 + 49478 T + p^{5} T^{2} \) |
| 67 | \( 1 + 59356 T + p^{5} T^{2} \) |
| 71 | \( 1 - 32040 T + p^{5} T^{2} \) |
| 73 | \( 1 - 61846 T + p^{5} T^{2} \) |
| 79 | \( 1 + 65776 T + p^{5} T^{2} \) |
| 83 | \( 1 + 40188 T + p^{5} T^{2} \) |
| 89 | \( 1 - 7974 T + p^{5} T^{2} \) |
| 97 | \( 1 - 143662 T + p^{5} T^{2} \) |
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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
−11.33691382444602661676796984865, −10.60096114619047474880460011031, −9.120639165452258990306953998291, −8.663423568224121446368503890948, −7.70532215906419380097637815362, −6.73864989978780196308920997253, −4.48494206107780944893104157030, −3.56105570507868716757861496148, −1.45458680556475004975894606522, 0,
1.45458680556475004975894606522, 3.56105570507868716757861496148, 4.48494206107780944893104157030, 6.73864989978780196308920997253, 7.70532215906419380097637815362, 8.663423568224121446368503890948, 9.120639165452258990306953998291, 10.60096114619047474880460011031, 11.33691382444602661676796984865
