| L(s) = 1 | + 0.139·2-s − 7.98·4-s − 14.6·5-s − 24.3·7-s − 2.22·8-s − 2.03·10-s − 60.6·11-s + 18.7·13-s − 3.38·14-s + 63.5·16-s − 85.1·17-s + 82.0·19-s + 116.·20-s − 8.43·22-s + 36.1·23-s + 88.3·25-s + 2.61·26-s + 194.·28-s + 81.4·29-s + 263.·31-s + 26.6·32-s − 11.8·34-s + 355.·35-s + 70.9·37-s + 11.4·38-s + 32.4·40-s − 152.·41-s + ⋯ |
| L(s) = 1 | + 0.0491·2-s − 0.997·4-s − 1.30·5-s − 1.31·7-s − 0.0982·8-s − 0.0642·10-s − 1.66·11-s + 0.400·13-s − 0.0645·14-s + 0.992·16-s − 1.21·17-s + 0.990·19-s + 1.30·20-s − 0.0817·22-s + 0.327·23-s + 0.707·25-s + 0.0197·26-s + 1.30·28-s + 0.521·29-s + 1.52·31-s + 0.147·32-s − 0.0597·34-s + 1.71·35-s + 0.315·37-s + 0.0487·38-s + 0.128·40-s − 0.581·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 241 | \( 1 + 241T \) |
| good | 2 | \( 1 - 0.139T + 8T^{2} \) |
| 5 | \( 1 + 14.6T + 125T^{2} \) |
| 7 | \( 1 + 24.3T + 343T^{2} \) |
| 11 | \( 1 + 60.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 18.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 85.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 82.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 36.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 81.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 263.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 70.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 152.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 149.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 117.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 239.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 201.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 632.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 441.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 889.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 989.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 501.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 749.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.47e3T + 9.12e5T^{2} \) |
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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
−8.380322929694202901815601472420, −7.67215736038120409082150566083, −6.88540711693063036129264695847, −5.89351294439205577915585360708, −4.94769012548142234126547009174, −4.26545275524690111452296723090, −3.35707704401581238719529962330, −2.75884261442314503186733569410, −0.70470200666354521861673873631, 0,
0.70470200666354521861673873631, 2.75884261442314503186733569410, 3.35707704401581238719529962330, 4.26545275524690111452296723090, 4.94769012548142234126547009174, 5.89351294439205577915585360708, 6.88540711693063036129264695847, 7.67215736038120409082150566083, 8.380322929694202901815601472420
