| L(s) = 1 | − 1.48·2-s + 0.193·4-s − 1.19·7-s + 2.67·8-s − 11-s − 0.806·13-s + 1.76·14-s − 4.35·16-s + 3.76·17-s − 5.35·19-s + 1.48·22-s + 4·23-s + 1.19·26-s − 0.231·28-s + 4.31·29-s + 0.962·31-s + 1.09·32-s − 5.58·34-s − 1.61·37-s + 7.92·38-s − 9.08·41-s − 4.41·43-s − 0.193·44-s − 5.92·46-s + 12.3·47-s − 5.57·49-s − 0.156·52-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.0969·4-s − 0.451·7-s + 0.945·8-s − 0.301·11-s − 0.223·13-s + 0.472·14-s − 1.08·16-s + 0.913·17-s − 1.22·19-s + 0.315·22-s + 0.834·23-s + 0.234·26-s − 0.0437·28-s + 0.800·29-s + 0.172·31-s + 0.193·32-s − 0.957·34-s − 0.265·37-s + 1.28·38-s − 1.41·41-s − 0.673·43-s − 0.0292·44-s − 0.873·46-s + 1.79·47-s − 0.796·49-s − 0.0216·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 1.48T + 2T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 13 | \( 1 + 0.806T + 13T^{2} \) |
| 17 | \( 1 - 3.76T + 17T^{2} \) |
| 19 | \( 1 + 5.35T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 4.31T + 29T^{2} \) |
| 31 | \( 1 - 0.962T + 31T^{2} \) |
| 37 | \( 1 + 1.61T + 37T^{2} \) |
| 41 | \( 1 + 9.08T + 41T^{2} \) |
| 43 | \( 1 + 4.41T + 43T^{2} \) |
| 47 | \( 1 - 12.3T + 47T^{2} \) |
| 53 | \( 1 - 1.42T + 53T^{2} \) |
| 59 | \( 1 + 13.2T + 59T^{2} \) |
| 61 | \( 1 + 0.0752T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 - 13.9T + 79T^{2} \) |
| 83 | \( 1 + 9.89T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−8.462905981897675273603091165955, −8.096948336931506763662053571992, −7.11438919197779585593948359412, −6.52848434501820639830704194590, −5.37768348106674159473634507493, −4.61343399479422484081707187362, −3.57674731730921617726390909834, −2.47222547831736259680970433414, −1.25610872776639772703473659592, 0,
1.25610872776639772703473659592, 2.47222547831736259680970433414, 3.57674731730921617726390909834, 4.61343399479422484081707187362, 5.37768348106674159473634507493, 6.52848434501820639830704194590, 7.11438919197779585593948359412, 8.096948336931506763662053571992, 8.462905981897675273603091165955
