L(s) = 1 | + 2-s − 3-s + 4-s − 0.137·5-s − 6-s + 7-s + 8-s + 9-s − 0.137·10-s − 1.48·11-s − 12-s + 5.15·13-s + 14-s + 0.137·15-s + 16-s − 6.22·17-s + 18-s + 4.44·19-s − 0.137·20-s − 21-s − 1.48·22-s − 0.508·23-s − 24-s − 4.98·25-s + 5.15·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.0612·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.0433·10-s − 0.447·11-s − 0.288·12-s + 1.42·13-s + 0.267·14-s + 0.0353·15-s + 0.250·16-s − 1.51·17-s + 0.235·18-s + 1.01·19-s − 0.0306·20-s − 0.218·21-s − 0.316·22-s − 0.105·23-s − 0.204·24-s − 0.996·25-s + 1.01·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
good | 5 | \( 1 + 0.137T + 5T^{2} \) |
| 11 | \( 1 + 1.48T + 11T^{2} \) |
| 13 | \( 1 - 5.15T + 13T^{2} \) |
| 17 | \( 1 + 6.22T + 17T^{2} \) |
| 19 | \( 1 - 4.44T + 19T^{2} \) |
| 23 | \( 1 + 0.508T + 23T^{2} \) |
| 29 | \( 1 + 6.50T + 29T^{2} \) |
| 31 | \( 1 + 5.10T + 31T^{2} \) |
| 37 | \( 1 + 5.93T + 37T^{2} \) |
| 41 | \( 1 + 5.58T + 41T^{2} \) |
| 43 | \( 1 - 3.67T + 43T^{2} \) |
| 47 | \( 1 + 10.1T + 47T^{2} \) |
| 53 | \( 1 + 4.64T + 53T^{2} \) |
| 59 | \( 1 + 5.05T + 59T^{2} \) |
| 61 | \( 1 - 5.53T + 61T^{2} \) |
| 67 | \( 1 + 5.75T + 67T^{2} \) |
| 71 | \( 1 - 9.62T + 71T^{2} \) |
| 73 | \( 1 + 7.16T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 + 4.16T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 - 0.295T + 97T^{2} \) |
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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
−7.38294838025711470406052588316, −6.61114768242712018718006321411, −6.03056962568308649993263789935, −5.34578995014024541339716898026, −4.80207099686751635273277046133, −3.86636123593745575254536969447, −3.41740669369837089465319676703, −2.13816091318936627453229162457, −1.46206943192069236130171091287, 0,
1.46206943192069236130171091287, 2.13816091318936627453229162457, 3.41740669369837089465319676703, 3.86636123593745575254536969447, 4.80207099686751635273277046133, 5.34578995014024541339716898026, 6.03056962568308649993263789935, 6.61114768242712018718006321411, 7.38294838025711470406052588316