| L(s) = 1 | − 3.47·2-s + 4.08·4-s − 7.60·5-s + 16.3·7-s + 13.6·8-s + 26.4·10-s − 42.4·11-s + 62.2·13-s − 56.9·14-s − 79.9·16-s + 17·17-s − 52.0·19-s − 31.0·20-s + 147.·22-s − 67.8·23-s − 67.1·25-s − 216.·26-s + 67.0·28-s − 91.7·29-s − 177.·31-s + 169.·32-s − 59.1·34-s − 124.·35-s − 258.·37-s + 181.·38-s − 103.·40-s − 262.·41-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.511·4-s − 0.680·5-s + 0.885·7-s + 0.601·8-s + 0.836·10-s − 1.16·11-s + 1.32·13-s − 1.08·14-s − 1.24·16-s + 0.242·17-s − 0.629·19-s − 0.347·20-s + 1.42·22-s − 0.615·23-s − 0.537·25-s − 1.63·26-s + 0.452·28-s − 0.587·29-s − 1.02·31-s + 0.935·32-s − 0.298·34-s − 0.602·35-s − 1.14·37-s + 0.773·38-s − 0.408·40-s − 0.998·41-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 | \( 1 + 3.47T + 8T^{2} \) |
| 5 | \( 1 + 7.60T + 125T^{2} \) |
| 7 | \( 1 - 16.3T + 343T^{2} \) |
| 11 | \( 1 + 42.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.2T + 2.19e3T^{2} \) |
| 19 | \( 1 + 52.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 67.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 91.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + 177.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 262.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 331.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 49.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 462.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 351.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 41.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 845.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 420.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 402.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 736.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.24e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 463.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−11.55547111339718698293511155345, −10.87122993994121819521119320620, −10.00466396333129140096534181900, −8.504311086719805608011508915068, −8.212111616199779670719115383464, −7.14072566283350107806256724454, −5.36979068699817131650604523531, −3.91901916700340289794050506332, −1.74777928361413249799629162791, 0,
1.74777928361413249799629162791, 3.91901916700340289794050506332, 5.36979068699817131650604523531, 7.14072566283350107806256724454, 8.212111616199779670719115383464, 8.504311086719805608011508915068, 10.00466396333129140096534181900, 10.87122993994121819521119320620, 11.55547111339718698293511155345
