| L(s) = 1 | − 0.364·2-s − 1.86·4-s − 0.715·7-s + 1.41·8-s − 3.71·11-s − 5.90·13-s + 0.260·14-s + 3.21·16-s − 6.46·17-s − 4.69·19-s + 1.35·22-s − 2.26·23-s + 2.15·26-s + 1.33·28-s − 9.36·29-s + 31-s − 3.99·32-s + 2.36·34-s − 8.82·37-s + 1.71·38-s + 2.31·41-s − 0.929·43-s + 6.93·44-s + 0.826·46-s + 8.18·47-s − 6.48·49-s + 11.0·52-s + ⋯ |
| L(s) = 1 | − 0.258·2-s − 0.933·4-s − 0.270·7-s + 0.498·8-s − 1.12·11-s − 1.63·13-s + 0.0697·14-s + 0.804·16-s − 1.56·17-s − 1.07·19-s + 0.289·22-s − 0.472·23-s + 0.422·26-s + 0.252·28-s − 1.73·29-s + 0.179·31-s − 0.706·32-s + 0.404·34-s − 1.45·37-s + 0.277·38-s + 0.361·41-s − 0.141·43-s + 1.04·44-s + 0.121·46-s + 1.19·47-s − 0.926·49-s + 1.52·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04004981253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04004981253\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.364T + 2T^{2} \) |
| 7 | \( 1 + 0.715T + 7T^{2} \) |
| 11 | \( 1 + 3.71T + 11T^{2} \) |
| 13 | \( 1 + 5.90T + 13T^{2} \) |
| 17 | \( 1 + 6.46T + 17T^{2} \) |
| 19 | \( 1 + 4.69T + 19T^{2} \) |
| 23 | \( 1 + 2.26T + 23T^{2} \) |
| 29 | \( 1 + 9.36T + 29T^{2} \) |
| 37 | \( 1 + 8.82T + 37T^{2} \) |
| 41 | \( 1 - 2.31T + 41T^{2} \) |
| 43 | \( 1 + 0.929T + 43T^{2} \) |
| 47 | \( 1 - 8.18T + 47T^{2} \) |
| 53 | \( 1 + 2.25T + 53T^{2} \) |
| 59 | \( 1 - 5.09T + 59T^{2} \) |
| 61 | \( 1 + 8.02T + 61T^{2} \) |
| 67 | \( 1 - 2.00T + 67T^{2} \) |
| 71 | \( 1 + 3.94T + 71T^{2} \) |
| 73 | \( 1 - 0.757T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 5.28T + 83T^{2} \) |
| 89 | \( 1 - 5.44T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−7.984057119784898683689135775575, −7.38266780488914681640234664729, −6.68307288592284943114553515716, −5.68848072412209616770441268665, −5.02858306035656145734827260911, −4.46727018506783824099370317088, −3.70805905541159833525436385121, −2.56163373351543565587145361778, −1.92918168382012197357599485901, −0.099379407369388268255190497758,
0.099379407369388268255190497758, 1.92918168382012197357599485901, 2.56163373351543565587145361778, 3.70805905541159833525436385121, 4.46727018506783824099370317088, 5.02858306035656145734827260911, 5.68848072412209616770441268665, 6.68307288592284943114553515716, 7.38266780488914681640234664729, 7.984057119784898683689135775575
