L(s) = 1 | + 4.56·2-s + 3·3-s + 12.8·4-s + 6.28·5-s + 13.7·6-s + 33.7·7-s + 22.2·8-s + 9·9-s + 28.6·10-s − 11·11-s + 38.6·12-s − 12.6·13-s + 154.·14-s + 18.8·15-s − 1.33·16-s + 59.1·17-s + 41.1·18-s + 3.73·19-s + 80.8·20-s + 101.·21-s − 50.2·22-s − 57.6·23-s + 66.7·24-s − 85.5·25-s − 57.8·26-s + 27·27-s + 434.·28-s + ⋯ |
L(s) = 1 | + 1.61·2-s + 0.577·3-s + 1.60·4-s + 0.561·5-s + 0.932·6-s + 1.82·7-s + 0.983·8-s + 0.333·9-s + 0.907·10-s − 0.301·11-s + 0.928·12-s − 0.270·13-s + 2.94·14-s + 0.324·15-s − 0.0208·16-s + 0.843·17-s + 0.538·18-s + 0.0451·19-s + 0.903·20-s + 1.05·21-s − 0.486·22-s − 0.522·23-s + 0.567·24-s − 0.684·25-s − 0.436·26-s + 0.192·27-s + 2.92·28-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
\(L(2)\) |
\(\approx\) |
\(10.61181711\) |
\(L(\frac12)\) |
\(\approx\) |
\(10.61181711\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.56T + 8T^{2} \) |
| 5 | \( 1 - 6.28T + 125T^{2} \) |
| 7 | \( 1 - 33.7T + 343T^{2} \) |
| 13 | \( 1 + 12.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 59.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 3.73T + 6.85e3T^{2} \) |
| 23 | \( 1 + 57.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 198.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 170.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 152.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 419.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 196.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 34.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 40.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 304.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 178.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 154.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 582.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 719.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.29e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 633.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 380.T + 9.12e5T^{2} \) |
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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
−8.554756613787743224983534515593, −7.893796416354783169342903336291, −7.19332966174001518398000831740, −6.03973289257256014377059311457, −5.44488983865918830248108864755, −4.64708888020615011664034170413, −4.13194795687427792435868617260, −2.90756385259993314905135165538, −2.21031042387779133589083895339, −1.29505891491471214635810389512,
1.29505891491471214635810389512, 2.21031042387779133589083895339, 2.90756385259993314905135165538, 4.13194795687427792435868617260, 4.64708888020615011664034170413, 5.44488983865918830248108864755, 6.03973289257256014377059311457, 7.19332966174001518398000831740, 7.893796416354783169342903336291, 8.554756613787743224983534515593