| L(s) = 1 | + 3.56·2-s + 4.72·4-s − 20.6·5-s − 5.24·7-s − 11.6·8-s − 73.6·10-s − 5.21·11-s − 14.8·13-s − 18.7·14-s − 79.4·16-s + 17·17-s + 26.2·19-s − 97.5·20-s − 18.5·22-s − 165.·23-s + 300.·25-s − 52.8·26-s − 24.7·28-s + 42.3·29-s + 263.·31-s − 190.·32-s + 60.6·34-s + 108.·35-s − 322.·37-s + 93.6·38-s + 240.·40-s − 321.·41-s + ⋯ |
| L(s) = 1 | + 1.26·2-s + 0.591·4-s − 1.84·5-s − 0.283·7-s − 0.515·8-s − 2.32·10-s − 0.142·11-s − 0.315·13-s − 0.357·14-s − 1.24·16-s + 0.242·17-s + 0.317·19-s − 1.09·20-s − 0.180·22-s − 1.49·23-s + 2.40·25-s − 0.398·26-s − 0.167·28-s + 0.271·29-s + 1.52·31-s − 1.05·32-s + 0.305·34-s + 0.522·35-s − 1.43·37-s + 0.399·38-s + 0.951·40-s − 1.22·41-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 | \( 1 - 3.56T + 8T^{2} \) |
| 5 | \( 1 + 20.6T + 125T^{2} \) |
| 7 | \( 1 + 5.24T + 343T^{2} \) |
| 11 | \( 1 + 5.21T + 1.33e3T^{2} \) |
| 13 | \( 1 + 14.8T + 2.19e3T^{2} \) |
| 19 | \( 1 - 26.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 165.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 42.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 263.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 322.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 321.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 385.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 309.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 192.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 587.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 241.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 205.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 933.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 869.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 102.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 298.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 666.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.35e3T + 9.12e5T^{2} \) |
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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
−11.96802574096708553715490528967, −11.68623690867923413808601613238, −10.21106716116437905220289577343, −8.631864082395563892075012110325, −7.64099022521358973174475380491, −6.45211447012361082084340485112, −4.96451583537262319485864484390, −4.01704296448586076519686293620, −3.08465486289132934211080619057, 0,
3.08465486289132934211080619057, 4.01704296448586076519686293620, 4.96451583537262319485864484390, 6.45211447012361082084340485112, 7.64099022521358973174475380491, 8.631864082395563892075012110325, 10.21106716116437905220289577343, 11.68623690867923413808601613238, 11.96802574096708553715490528967
