| L(s) = 1 | + 3.73·2-s + 5.94·4-s − 7·7-s − 7.67·8-s − 36.3·11-s + 75.8·13-s − 26.1·14-s − 76.2·16-s + 31.6·17-s + 25.1·19-s − 135.·22-s + 212.·23-s + 283.·26-s − 41.6·28-s − 235.·29-s − 270.·31-s − 223.·32-s + 118.·34-s − 362.·37-s + 93.9·38-s + 132.·41-s + 5.13·43-s − 216.·44-s + 792.·46-s + 216.·47-s + 49·49-s + 451.·52-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.743·4-s − 0.377·7-s − 0.338·8-s − 0.996·11-s + 1.61·13-s − 0.499·14-s − 1.19·16-s + 0.451·17-s + 0.303·19-s − 1.31·22-s + 1.92·23-s + 2.13·26-s − 0.280·28-s − 1.50·29-s − 1.56·31-s − 1.23·32-s + 0.596·34-s − 1.60·37-s + 0.400·38-s + 0.505·41-s + 0.0182·43-s − 0.740·44-s + 2.54·46-s + 0.672·47-s + 0.142·49-s + 1.20·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 3.73T + 8T^{2} \) |
| 11 | \( 1 + 36.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 75.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 25.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 212.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 270.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 362.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 132.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 5.13T + 7.95e4T^{2} \) |
| 47 | \( 1 - 216.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 455.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 689.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 130.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 633.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 381.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 48.5T + 5.71e5T^{2} \) |
| 89 | \( 1 - 53.5T + 7.04e5T^{2} \) |
| 97 | \( 1 + 968.T + 9.12e5T^{2} \) |
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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
−8.864097014813952614712506988965, −7.60965437775468333573537026438, −6.88770131615271675888920221165, −5.76478812470396879062321937147, −5.51355889518387741700499709036, −4.46185867716856648885859200228, −3.43260672023237240532966178291, −3.04053000625369225000075283876, −1.56783234319818570626374882233, 0,
1.56783234319818570626374882233, 3.04053000625369225000075283876, 3.43260672023237240532966178291, 4.46185867716856648885859200228, 5.51355889518387741700499709036, 5.76478812470396879062321937147, 6.88770131615271675888920221165, 7.60965437775468333573537026438, 8.864097014813952614712506988965
