L(s) = 1 | − 2.21·2-s − 3·3-s − 3.08·4-s − 20.1·5-s + 6.64·6-s + 13.2·7-s + 24.5·8-s + 9·9-s + 44.6·10-s − 11·11-s + 9.26·12-s + 59.5·13-s − 29.2·14-s + 60.4·15-s − 29.7·16-s + 67.8·17-s − 19.9·18-s + 19.2·19-s + 62.2·20-s − 39.6·21-s + 24.3·22-s + 112.·23-s − 73.7·24-s + 281.·25-s − 131.·26-s − 27·27-s − 40.8·28-s + ⋯ |
L(s) = 1 | − 0.783·2-s − 0.577·3-s − 0.385·4-s − 1.80·5-s + 0.452·6-s + 0.713·7-s + 1.08·8-s + 0.333·9-s + 1.41·10-s − 0.301·11-s + 0.222·12-s + 1.27·13-s − 0.559·14-s + 1.04·15-s − 0.465·16-s + 0.968·17-s − 0.261·18-s + 0.232·19-s + 0.696·20-s − 0.411·21-s + 0.236·22-s + 1.01·23-s − 0.627·24-s + 2.25·25-s − 0.995·26-s − 0.192·27-s − 0.275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.055678917\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.055678917\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
good | 2 | \( 1 + 2.21T + 8T^{2} \) |
| 5 | \( 1 + 20.1T + 125T^{2} \) |
| 7 | \( 1 - 13.2T + 343T^{2} \) |
| 13 | \( 1 - 59.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 19.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 235.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 336.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 130.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 466.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 71.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 390.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 19.9T + 2.05e5T^{2} \) |
| 67 | \( 1 + 206.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 206.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 757.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 118.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 613.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 434.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 981.T + 9.12e5T^{2} \) |
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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
−8.534049199636678277325850111132, −8.053735371473422500007308288074, −7.61506214356497439426958081427, −6.69003109781943394427250288116, −5.48345487842022665934324097171, −4.54342597944597906181598263151, −4.12311627119401459193111221668, −3.00431067630835784816297982063, −0.968458871822237505014973207817, −0.823852243193761481877916519310,
0.823852243193761481877916519310, 0.968458871822237505014973207817, 3.00431067630835784816297982063, 4.12311627119401459193111221668, 4.54342597944597906181598263151, 5.48345487842022665934324097171, 6.69003109781943394427250288116, 7.61506214356497439426958081427, 8.053735371473422500007308288074, 8.534049199636678277325850111132