| L(s) = 1 | − 2-s + 4-s + 5-s − 1.31·7-s − 8-s − 10-s + 0.258·11-s − 5.87·13-s + 1.31·14-s + 16-s − 4.25·17-s + 2.93·19-s + 20-s − 0.258·22-s + 8.51·23-s + 25-s + 5.87·26-s − 1.31·28-s + 3.61·29-s + 3.95·31-s − 32-s + 4.25·34-s − 1.31·35-s − 37-s − 2.93·38-s − 40-s + 8.89·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.498·7-s − 0.353·8-s − 0.316·10-s + 0.0780·11-s − 1.63·13-s + 0.352·14-s + 0.250·16-s − 1.03·17-s + 0.674·19-s + 0.223·20-s − 0.0551·22-s + 1.77·23-s + 0.200·25-s + 1.15·26-s − 0.249·28-s + 0.672·29-s + 0.710·31-s − 0.176·32-s + 0.730·34-s − 0.223·35-s − 0.164·37-s − 0.476·38-s − 0.158·40-s + 1.38·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 1.31T + 7T^{2} \) |
| 11 | \( 1 - 0.258T + 11T^{2} \) |
| 13 | \( 1 + 5.87T + 13T^{2} \) |
| 17 | \( 1 + 4.25T + 17T^{2} \) |
| 19 | \( 1 - 2.93T + 19T^{2} \) |
| 23 | \( 1 - 8.51T + 23T^{2} \) |
| 29 | \( 1 - 3.61T + 29T^{2} \) |
| 31 | \( 1 - 3.95T + 31T^{2} \) |
| 41 | \( 1 - 8.89T + 41T^{2} \) |
| 43 | \( 1 + 5.61T + 43T^{2} \) |
| 47 | \( 1 + 5.57T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 9.57T + 59T^{2} \) |
| 61 | \( 1 - 0.380T + 61T^{2} \) |
| 67 | \( 1 - 5.57T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 2.51T + 73T^{2} \) |
| 79 | \( 1 + 7.69T + 79T^{2} \) |
| 83 | \( 1 - 6.17T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 12.8T + 97T^{2} \) |
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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
−8.319740252622533988771028010857, −7.47712137994913074806558986733, −6.82721943116203372571817642309, −6.28372532299509941393636028802, −5.14353736182402394092052018844, −4.58545706513461456089084257968, −3.08201735038484098590299182150, −2.58341573525143765541493785370, −1.36850503802492192231121032476, 0,
1.36850503802492192231121032476, 2.58341573525143765541493785370, 3.08201735038484098590299182150, 4.58545706513461456089084257968, 5.14353736182402394092052018844, 6.28372532299509941393636028802, 6.82721943116203372571817642309, 7.47712137994913074806558986733, 8.319740252622533988771028010857
