| L(s) = 1 | + 4.38·2-s + 11.2·4-s − 5·5-s − 11.7·7-s + 14.2·8-s − 21.9·10-s − 11·11-s − 72.8·13-s − 51.4·14-s − 27.3·16-s + 9.89·17-s + 0.0238·19-s − 56.2·20-s − 48.2·22-s − 73.0·23-s + 25·25-s − 319.·26-s − 132.·28-s − 202.·29-s + 181.·31-s − 234.·32-s + 43.4·34-s + 58.6·35-s + 299.·37-s + 0.104·38-s − 71.4·40-s − 88.5·41-s + ⋯ |
| L(s) = 1 | + 1.55·2-s + 1.40·4-s − 0.447·5-s − 0.633·7-s + 0.631·8-s − 0.693·10-s − 0.301·11-s − 1.55·13-s − 0.982·14-s − 0.426·16-s + 0.141·17-s + 0.000288·19-s − 0.629·20-s − 0.467·22-s − 0.662·23-s + 0.200·25-s − 2.41·26-s − 0.891·28-s − 1.29·29-s + 1.05·31-s − 1.29·32-s + 0.219·34-s + 0.283·35-s + 1.33·37-s + 0.000446·38-s − 0.282·40-s − 0.337·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 4.38T + 8T^{2} \) |
| 7 | \( 1 + 11.7T + 343T^{2} \) |
| 13 | \( 1 + 72.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.89T + 4.91e3T^{2} \) |
| 19 | \( 1 - 0.0238T + 6.85e3T^{2} \) |
| 23 | \( 1 + 73.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 202.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 181.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 299.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 88.5T + 6.89e4T^{2} \) |
| 43 | \( 1 + 146.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 185.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 347.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 691.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 491.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 715.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 541.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 159.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 212.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 413.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 567.T + 9.12e5T^{2} \) |
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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
−10.19814510499849818653376349681, −9.387555405987091641523616478851, −7.960236958444433142419406646491, −7.07207213718760181131703582434, −6.14915703521124997715990092099, −5.15823084582792951059069945294, −4.32709804378614209444557054314, −3.28850879807288255348413513510, −2.34391271702565884038449679734, 0,
2.34391271702565884038449679734, 3.28850879807288255348413513510, 4.32709804378614209444557054314, 5.15823084582792951059069945294, 6.14915703521124997715990092099, 7.07207213718760181131703582434, 7.960236958444433142419406646491, 9.387555405987091641523616478851, 10.19814510499849818653376349681
