L(s) = 1 | + 2-s + 1.77·3-s + 4-s − 0.491·5-s + 1.77·6-s + 4.65·7-s + 8-s + 0.145·9-s − 0.491·10-s + 3.35·11-s + 1.77·12-s + 4.65·14-s − 0.872·15-s + 16-s − 3.37·17-s + 0.145·18-s + 19-s − 0.491·20-s + 8.26·21-s + 3.35·22-s + 3.53·23-s + 1.77·24-s − 4.75·25-s − 5.06·27-s + 4.65·28-s − 0.803·29-s − 0.872·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.02·3-s + 0.5·4-s − 0.219·5-s + 0.724·6-s + 1.76·7-s + 0.353·8-s + 0.0486·9-s − 0.155·10-s + 1.01·11-s + 0.512·12-s + 1.24·14-s − 0.225·15-s + 0.250·16-s − 0.819·17-s + 0.0343·18-s + 0.229·19-s − 0.109·20-s + 1.80·21-s + 0.715·22-s + 0.737·23-s + 0.362·24-s − 0.951·25-s − 0.974·27-s + 0.880·28-s − 0.149·29-s − 0.159·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.918718958\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.918718958\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 1.77T + 3T^{2} \) |
| 5 | \( 1 + 0.491T + 5T^{2} \) |
| 7 | \( 1 - 4.65T + 7T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 17 | \( 1 + 3.37T + 17T^{2} \) |
| 23 | \( 1 - 3.53T + 23T^{2} \) |
| 29 | \( 1 + 0.803T + 29T^{2} \) |
| 31 | \( 1 - 9.31T + 31T^{2} \) |
| 37 | \( 1 + 7.90T + 37T^{2} \) |
| 41 | \( 1 + 5.41T + 41T^{2} \) |
| 43 | \( 1 - 7.27T + 43T^{2} \) |
| 47 | \( 1 - 7.21T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 5.74T + 59T^{2} \) |
| 61 | \( 1 - 9.22T + 61T^{2} \) |
| 67 | \( 1 - 3.79T + 67T^{2} \) |
| 71 | \( 1 - 8.43T + 71T^{2} \) |
| 73 | \( 1 + 1.63T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 + 8.43T + 97T^{2} \) |
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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
−8.130773310208828199535396792072, −7.34027364969213019173338390910, −6.75791793317202066120032905625, −5.70960319466055408534307985574, −5.05418756991731450864980763026, −4.19785065760872176646625106541, −3.83669406293597289526230940399, −2.71614236828118323630135048154, −2.05737234482029047602580386788, −1.19272421327097452460447221798,
1.19272421327097452460447221798, 2.05737234482029047602580386788, 2.71614236828118323630135048154, 3.83669406293597289526230940399, 4.19785065760872176646625106541, 5.05418756991731450864980763026, 5.70960319466055408534307985574, 6.75791793317202066120032905625, 7.34027364969213019173338390910, 8.130773310208828199535396792072