| L(s) = 1 | + 9.21·2-s + 52.9·4-s − 167.·7-s + 193.·8-s − 593.·11-s − 251.·13-s − 1.54e3·14-s + 85.5·16-s − 372.·17-s − 1.00e3·19-s − 5.46e3·22-s + 2.53e3·23-s − 2.32e3·26-s − 8.85e3·28-s + 5.85e3·29-s − 6.50e3·31-s − 5.39e3·32-s − 3.43e3·34-s − 5.47e3·37-s − 9.28e3·38-s + 1.33e4·41-s − 2.55e3·43-s − 3.14e4·44-s + 2.33e4·46-s + 7.69e3·47-s + 1.11e4·49-s − 1.33e4·52-s + ⋯ |
| L(s) = 1 | + 1.62·2-s + 1.65·4-s − 1.28·7-s + 1.06·8-s − 1.47·11-s − 0.413·13-s − 2.10·14-s + 0.0835·16-s − 0.312·17-s − 0.640·19-s − 2.40·22-s + 0.999·23-s − 0.673·26-s − 2.13·28-s + 1.29·29-s − 1.21·31-s − 0.930·32-s − 0.509·34-s − 0.657·37-s − 1.04·38-s + 1.24·41-s − 0.211·43-s − 2.44·44-s + 1.62·46-s + 0.508·47-s + 0.663·49-s − 0.683·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 9.21T + 32T^{2} \) |
| 7 | \( 1 + 167.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 593.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 251.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 372.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.00e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.53e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.50e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.47e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.33e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.55e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.69e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.59e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.28e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.00e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.00e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.47e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.59e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.08e4T + 8.58e9T^{2} \) |
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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
−11.07050330239817598146237762777, −10.24578492055392414691644532616, −8.970424239318706673926161403933, −7.40937732158371478272968321438, −6.48549105332996040294271121706, −5.52051416122668004953257340565, −4.54192644970125128986069695296, −3.24177029129062353982154712161, −2.47547932227067708239122647207, 0,
2.47547932227067708239122647207, 3.24177029129062353982154712161, 4.54192644970125128986069695296, 5.52051416122668004953257340565, 6.48549105332996040294271121706, 7.40937732158371478272968321438, 8.970424239318706673926161403933, 10.24578492055392414691644532616, 11.07050330239817598146237762777
