L(s) = 1 | − 2-s − 1.95·3-s + 4-s − 1.60·5-s + 1.95·6-s − 2.46·7-s − 8-s + 0.811·9-s + 1.60·10-s − 1.22·11-s − 1.95·12-s − 0.0707·13-s + 2.46·14-s + 3.12·15-s + 16-s − 0.614·17-s − 0.811·18-s − 4.27·19-s − 1.60·20-s + 4.80·21-s + 1.22·22-s + 3.57·23-s + 1.95·24-s − 2.43·25-s + 0.0707·26-s + 4.27·27-s − 2.46·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.12·3-s + 0.5·4-s − 0.716·5-s + 0.797·6-s − 0.930·7-s − 0.353·8-s + 0.270·9-s + 0.506·10-s − 0.368·11-s − 0.563·12-s − 0.0196·13-s + 0.658·14-s + 0.808·15-s + 0.250·16-s − 0.149·17-s − 0.191·18-s − 0.979·19-s − 0.358·20-s + 1.04·21-s + 0.260·22-s + 0.744·23-s + 0.398·24-s − 0.486·25-s + 0.0138·26-s + 0.822·27-s − 0.465·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 + 1.60T + 5T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 + 1.22T + 11T^{2} \) |
| 13 | \( 1 + 0.0707T + 13T^{2} \) |
| 17 | \( 1 + 0.614T + 17T^{2} \) |
| 19 | \( 1 + 4.27T + 19T^{2} \) |
| 23 | \( 1 - 3.57T + 23T^{2} \) |
| 29 | \( 1 - 6.60T + 29T^{2} \) |
| 31 | \( 1 - 6.99T + 31T^{2} \) |
| 37 | \( 1 - 3.78T + 37T^{2} \) |
| 41 | \( 1 + 7.69T + 41T^{2} \) |
| 43 | \( 1 + 2.94T + 43T^{2} \) |
| 47 | \( 1 - 4.45T + 47T^{2} \) |
| 53 | \( 1 - 5.25T + 53T^{2} \) |
| 59 | \( 1 + 1.04T + 59T^{2} \) |
| 61 | \( 1 - 6.55T + 61T^{2} \) |
| 67 | \( 1 - 0.787T + 67T^{2} \) |
| 71 | \( 1 - 0.133T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 3.58T + 79T^{2} \) |
| 83 | \( 1 + 9.60T + 83T^{2} \) |
| 89 | \( 1 + 6.25T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−8.244259352178903819958034204244, −7.21684201030005110801645302821, −6.56611080751284129027126411658, −6.14279215447819736039348718477, −5.16706754145501722406157170740, −4.37105518414998373593752859566, −3.31378057643953208220252829952, −2.44014912635091964796370024498, −0.863945768757779416634012496329, 0,
0.863945768757779416634012496329, 2.44014912635091964796370024498, 3.31378057643953208220252829952, 4.37105518414998373593752859566, 5.16706754145501722406157170740, 6.14279215447819736039348718477, 6.56611080751284129027126411658, 7.21684201030005110801645302821, 8.244259352178903819958034204244