L(s) = 1 | + 2-s + 2.47·3-s + 4-s + 0.657·5-s + 2.47·6-s − 1.67·7-s + 8-s + 3.11·9-s + 0.657·10-s + 4.79·11-s + 2.47·12-s − 1.67·14-s + 1.62·15-s + 16-s − 1.91·17-s + 3.11·18-s + 19-s + 0.657·20-s − 4.12·21-s + 4.79·22-s + 4.96·23-s + 2.47·24-s − 4.56·25-s + 0.273·27-s − 1.67·28-s + 6.18·29-s + 1.62·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.42·3-s + 0.5·4-s + 0.294·5-s + 1.00·6-s − 0.631·7-s + 0.353·8-s + 1.03·9-s + 0.207·10-s + 1.44·11-s + 0.713·12-s − 0.446·14-s + 0.419·15-s + 0.250·16-s − 0.465·17-s + 0.733·18-s + 0.229·19-s + 0.147·20-s − 0.900·21-s + 1.02·22-s + 1.03·23-s + 0.504·24-s − 0.913·25-s + 0.0525·27-s − 0.315·28-s + 1.14·29-s + 0.296·30-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
\(L(1)\) |
\(\approx\) |
\(6.136517288\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.136517288\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.47T + 3T^{2} \) |
| 5 | \( 1 - 0.657T + 5T^{2} \) |
| 7 | \( 1 + 1.67T + 7T^{2} \) |
| 11 | \( 1 - 4.79T + 11T^{2} \) |
| 17 | \( 1 + 1.91T + 17T^{2} \) |
| 23 | \( 1 - 4.96T + 23T^{2} \) |
| 29 | \( 1 - 6.18T + 29T^{2} \) |
| 31 | \( 1 - 5.42T + 31T^{2} \) |
| 37 | \( 1 + 0.670T + 37T^{2} \) |
| 41 | \( 1 - 5.58T + 41T^{2} \) |
| 43 | \( 1 + 7.81T + 43T^{2} \) |
| 47 | \( 1 - 1.70T + 47T^{2} \) |
| 53 | \( 1 - 4.47T + 53T^{2} \) |
| 59 | \( 1 - 1.28T + 59T^{2} \) |
| 61 | \( 1 - 0.0985T + 61T^{2} \) |
| 67 | \( 1 - 9.67T + 67T^{2} \) |
| 71 | \( 1 + 8.55T + 71T^{2} \) |
| 73 | \( 1 + 5.77T + 73T^{2} \) |
| 79 | \( 1 - 7.95T + 79T^{2} \) |
| 83 | \( 1 - 1.72T + 83T^{2} \) |
| 89 | \( 1 - 9.87T + 89T^{2} \) |
| 97 | \( 1 - 9.59T + 97T^{2} \) |
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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
−8.077744169538190993420111518438, −7.21487239570442246810282213683, −6.58906267761408992616398544854, −6.09239874243517248950786504225, −4.97431098211210237939501235565, −4.16748586296761855389805165233, −3.54198835975428691523562249034, −2.90247031701052661407702350210, −2.15416365606706705669117637708, −1.16321342482419311914367702722,
1.16321342482419311914367702722, 2.15416365606706705669117637708, 2.90247031701052661407702350210, 3.54198835975428691523562249034, 4.16748586296761855389805165233, 4.97431098211210237939501235565, 6.09239874243517248950786504225, 6.58906267761408992616398544854, 7.21487239570442246810282213683, 8.077744169538190993420111518438