| L(s) = 1 | − 3-s + 2·5-s + 9-s − 6·13-s − 2·15-s − 6·17-s + 19-s + 4·23-s − 25-s − 27-s − 2·29-s + 8·31-s + 10·37-s + 6·39-s − 2·41-s + 4·43-s + 2·45-s + 12·47-s − 7·49-s + 6·51-s + 6·53-s − 57-s + 12·59-s + 2·61-s − 12·65-s + 4·67-s − 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s − 1.66·13-s − 0.516·15-s − 1.45·17-s + 0.229·19-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s + 1.64·37-s + 0.960·39-s − 0.312·41-s + 0.609·43-s + 0.298·45-s + 1.75·47-s − 49-s + 0.840·51-s + 0.824·53-s − 0.132·57-s + 1.56·59-s + 0.256·61-s − 1.48·65-s + 0.488·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.534849192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534849192\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.656147919084617168059664238609, −7.62692916507233066692617069633, −6.96313967889007715870809714116, −6.30766736214003560110551425788, −5.52157301147752958888078955745, −4.83501817913468223446739446530, −4.16209472915564587197687363565, −2.66586826806470965226318272179, −2.15027495389545596722403606493, −0.72964391205275968179204890320,
0.72964391205275968179204890320, 2.15027495389545596722403606493, 2.66586826806470965226318272179, 4.16209472915564587197687363565, 4.83501817913468223446739446530, 5.52157301147752958888078955745, 6.30766736214003560110551425788, 6.96313967889007715870809714116, 7.62692916507233066692617069633, 8.656147919084617168059664238609
