| L(s) = 1 | + 2.55·2-s − 1.47·4-s − 8.47·5-s + 3.66·7-s − 24.1·8-s − 21.6·10-s − 61.4·11-s + 20.9·13-s + 9.35·14-s − 50.0·16-s − 17·17-s − 102.·19-s + 12.4·20-s − 157.·22-s + 27.1·23-s − 53.2·25-s + 53.6·26-s − 5.38·28-s + 145.·29-s + 72.0·31-s + 65.6·32-s − 43.4·34-s − 31.0·35-s + 371.·37-s − 262.·38-s + 204.·40-s − 348.·41-s + ⋯ |
| L(s) = 1 | + 0.903·2-s − 0.183·4-s − 0.757·5-s + 0.197·7-s − 1.06·8-s − 0.684·10-s − 1.68·11-s + 0.447·13-s + 0.178·14-s − 0.782·16-s − 0.242·17-s − 1.24·19-s + 0.139·20-s − 1.52·22-s + 0.246·23-s − 0.425·25-s + 0.404·26-s − 0.0363·28-s + 0.930·29-s + 0.417·31-s + 0.362·32-s − 0.219·34-s − 0.149·35-s + 1.64·37-s − 1.12·38-s + 0.810·40-s − 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 - 2.55T + 8T^{2} \) |
| 5 | \( 1 + 8.47T + 125T^{2} \) |
| 7 | \( 1 - 3.66T + 343T^{2} \) |
| 11 | \( 1 + 61.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.9T + 2.19e3T^{2} \) |
| 19 | \( 1 + 102.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 27.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 145.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 72.0T + 2.97e4T^{2} \) |
| 37 | \( 1 - 371.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 348.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 246.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 269.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 349.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 78.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 410.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 493.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 480.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 524.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 189.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.04e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 725.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.75e3T + 9.12e5T^{2} \) |
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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
−12.23544764524275146374043854873, −11.21163196919064666561215266294, −10.16420266429647573781039250093, −8.644243741732146534086964003420, −7.88229179255241623027027399994, −6.32754042281170777383415357055, −5.08004569547810011465154599049, −4.14971281380827279224524226351, −2.77838577401269894070988374506, 0,
2.77838577401269894070988374506, 4.14971281380827279224524226351, 5.08004569547810011465154599049, 6.32754042281170777383415357055, 7.88229179255241623027027399994, 8.644243741732146534086964003420, 10.16420266429647573781039250093, 11.21163196919064666561215266294, 12.23544764524275146374043854873
