| L(s) = 1 | − 2-s + 4-s − 2.82·5-s − 8-s + 2.82·10-s + 2.82·11-s + 2·13-s + 16-s + 4·19-s − 2.82·20-s − 2.82·22-s − 5.65·23-s + 3.00·25-s − 2·26-s − 2.82·29-s − 32-s + 8.48·37-s − 4·38-s + 2.82·40-s − 5.65·41-s + 4·43-s + 2.82·44-s + 5.65·46-s − 7·49-s − 3.00·50-s + 2·52-s + 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.26·5-s − 0.353·8-s + 0.894·10-s + 0.852·11-s + 0.554·13-s + 0.250·16-s + 0.917·19-s − 0.632·20-s − 0.603·22-s − 1.17·23-s + 0.600·25-s − 0.392·26-s − 0.525·29-s − 0.176·32-s + 1.39·37-s − 0.648·38-s + 0.447·40-s − 0.883·41-s + 0.609·43-s + 0.426·44-s + 0.834·46-s − 49-s − 0.424·50-s + 0.277·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9888966443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9888966443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−8.295945313715230012877955557616, −7.55254775179995784928698343008, −7.03297809891797224280566885011, −6.20234574891670124970550698017, −5.43281390722808584454115403820, −4.19107148169896089870118935212, −3.81196431311926348108275062389, −2.87366943266412262528728859835, −1.64064148924853301281107604475, −0.61892610321797346724490240892,
0.61892610321797346724490240892, 1.64064148924853301281107604475, 2.87366943266412262528728859835, 3.81196431311926348108275062389, 4.19107148169896089870118935212, 5.43281390722808584454115403820, 6.20234574891670124970550698017, 7.03297809891797224280566885011, 7.55254775179995784928698343008, 8.295945313715230012877955557616
