| L(s) = 1 | − 4.85·2-s − 5.73·3-s + 15.6·4-s + 14.4·5-s + 27.8·6-s − 13.1·7-s − 36.9·8-s + 5.83·9-s − 70.2·10-s − 56.6·11-s − 89.4·12-s − 68.5·13-s + 64.0·14-s − 82.8·15-s + 54.6·16-s + 23.3·17-s − 28.3·18-s − 119.·19-s + 225.·20-s + 75.5·21-s + 275.·22-s − 49.8·23-s + 211.·24-s + 84.0·25-s + 333.·26-s + 121.·27-s − 205.·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 1.10·3-s + 1.95·4-s + 1.29·5-s + 1.89·6-s − 0.711·7-s − 1.63·8-s + 0.216·9-s − 2.22·10-s − 1.55·11-s − 2.15·12-s − 1.46·13-s + 1.22·14-s − 1.42·15-s + 0.854·16-s + 0.333·17-s − 0.371·18-s − 1.44·19-s + 2.52·20-s + 0.785·21-s + 2.66·22-s − 0.451·23-s + 1.80·24-s + 0.672·25-s + 2.51·26-s + 0.864·27-s − 1.38·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 4.85T + 8T^{2} \) |
| 3 | \( 1 + 5.73T + 27T^{2} \) |
| 5 | \( 1 - 14.4T + 125T^{2} \) |
| 7 | \( 1 + 13.1T + 343T^{2} \) |
| 11 | \( 1 + 56.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 68.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 23.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 119.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 49.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 290.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 58.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 147.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 216.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 233.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 712.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 70.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 328.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 307.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.13e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 203.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 255.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 665.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 834.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 817.T + 9.12e5T^{2} \) |
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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
−8.526202283828307679037679336255, −7.88764782848712559633596191825, −6.80458645042228078414721689677, −6.36760930326932259365738129419, −5.55306892885260627510543812949, −4.77512763780250061682941365325, −2.58132158635269619740504580913, −2.31926382920898895344640621249, −0.77813855566844007094210072974, 0,
0.77813855566844007094210072974, 2.31926382920898895344640621249, 2.58132158635269619740504580913, 4.77512763780250061682941365325, 5.55306892885260627510543812949, 6.36760930326932259365738129419, 6.80458645042228078414721689677, 7.88764782848712559633596191825, 8.526202283828307679037679336255
