| L(s) = 1 | − 3·3-s − 5·5-s − 7·7-s + 9·9-s − 54·13-s + 15·15-s + 74·17-s − 20·19-s + 21·21-s − 160·23-s + 25·25-s − 27·27-s − 246·29-s + 84·31-s + 35·35-s + 306·37-s + 162·39-s − 370·41-s − 88·43-s − 45·45-s + 460·47-s + 49·49-s − 222·51-s + 686·53-s + 60·57-s − 684·59-s + 186·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.15·13-s + 0.258·15-s + 1.05·17-s − 0.241·19-s + 0.218·21-s − 1.45·23-s + 1/5·25-s − 0.192·27-s − 1.57·29-s + 0.486·31-s + 0.169·35-s + 1.35·37-s + 0.665·39-s − 1.40·41-s − 0.312·43-s − 0.149·45-s + 1.42·47-s + 1/7·49-s − 0.609·51-s + 1.77·53-s + 0.139·57-s − 1.50·59-s + 0.390·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9426931048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9426931048\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 160 T + p^{3} T^{2} \) |
| 29 | \( 1 + 246 T + p^{3} T^{2} \) |
| 31 | \( 1 - 84 T + p^{3} T^{2} \) |
| 37 | \( 1 - 306 T + p^{3} T^{2} \) |
| 41 | \( 1 + 370 T + p^{3} T^{2} \) |
| 43 | \( 1 + 88 T + p^{3} T^{2} \) |
| 47 | \( 1 - 460 T + p^{3} T^{2} \) |
| 53 | \( 1 - 686 T + p^{3} T^{2} \) |
| 59 | \( 1 + 684 T + p^{3} T^{2} \) |
| 61 | \( 1 - 186 T + p^{3} T^{2} \) |
| 67 | \( 1 + 904 T + p^{3} T^{2} \) |
| 71 | \( 1 - 912 T + p^{3} T^{2} \) |
| 73 | \( 1 + 26 T + p^{3} T^{2} \) |
| 79 | \( 1 + 320 T + p^{3} T^{2} \) |
| 83 | \( 1 - 732 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1150 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1526 T + p^{3} T^{2} \) |
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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
−9.973865167940610734689336542021, −9.099693727737842741262319618358, −7.85563298689590552270145868120, −7.36414877579401367892080783573, −6.25310100907330996962786929381, −5.46061089489792237989369606763, −4.43188189905304208569554183276, −3.46412967930519860522546400117, −2.12080826408007260461578631576, −0.53562169839904021585016562966,
0.53562169839904021585016562966, 2.12080826408007260461578631576, 3.46412967930519860522546400117, 4.43188189905304208569554183276, 5.46061089489792237989369606763, 6.25310100907330996962786929381, 7.36414877579401367892080783573, 7.85563298689590552270145868120, 9.099693727737842741262319618358, 9.973865167940610734689336542021
