L(s) = 1 | − 2-s + 3-s + 4-s + 1.96·5-s − 6-s − 3.61·7-s − 8-s + 9-s − 1.96·10-s + 0.322·11-s + 12-s + 2.69·13-s + 3.61·14-s + 1.96·15-s + 16-s − 17-s − 18-s − 3.90·19-s + 1.96·20-s − 3.61·21-s − 0.322·22-s − 4.79·23-s − 24-s − 1.12·25-s − 2.69·26-s + 27-s − 3.61·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.879·5-s − 0.408·6-s − 1.36·7-s − 0.353·8-s + 0.333·9-s − 0.622·10-s + 0.0971·11-s + 0.288·12-s + 0.748·13-s + 0.966·14-s + 0.507·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 0.896·19-s + 0.439·20-s − 0.789·21-s − 0.0686·22-s − 0.999·23-s − 0.204·24-s − 0.225·25-s − 0.528·26-s + 0.192·27-s − 0.683·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.656425489\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.656425489\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
good | 5 | \( 1 - 1.96T + 5T^{2} \) |
| 7 | \( 1 + 3.61T + 7T^{2} \) |
| 11 | \( 1 - 0.322T + 11T^{2} \) |
| 13 | \( 1 - 2.69T + 13T^{2} \) |
| 19 | \( 1 + 3.90T + 19T^{2} \) |
| 23 | \( 1 + 4.79T + 23T^{2} \) |
| 29 | \( 1 + 0.708T + 29T^{2} \) |
| 31 | \( 1 - 6.38T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 - 1.55T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 12.0T + 47T^{2} \) |
| 53 | \( 1 + 1.01T + 53T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 - 12.2T + 67T^{2} \) |
| 71 | \( 1 - 7.07T + 71T^{2} \) |
| 73 | \( 1 - 3.18T + 73T^{2} \) |
| 79 | \( 1 + 0.0479T + 79T^{2} \) |
| 83 | \( 1 - 0.102T + 83T^{2} \) |
| 89 | \( 1 - 3.15T + 89T^{2} \) |
| 97 | \( 1 - 2.11T + 97T^{2} \) |
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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
−8.300354090189132048580733820081, −7.39717643354714115787234663929, −6.58171421742050750902427484028, −6.21528970363499698465289635524, −5.52658004436616627410384149511, −4.15312137529228180420744009098, −3.53509343352594899718176506426, −2.52315163885319717623102513985, −1.99400511970926957029571025385, −0.71196915385092737824283181565,
0.71196915385092737824283181565, 1.99400511970926957029571025385, 2.52315163885319717623102513985, 3.53509343352594899718176506426, 4.15312137529228180420744009098, 5.52658004436616627410384149511, 6.21528970363499698465289635524, 6.58171421742050750902427484028, 7.39717643354714115787234663929, 8.300354090189132048580733820081