L(s) = 1 | + 3.23·2-s + 3·3-s + 2.46·4-s + 19.2·5-s + 9.70·6-s + 32.7·7-s − 17.9·8-s + 9·9-s + 62.4·10-s − 11·11-s + 7.40·12-s + 24.7·13-s + 105.·14-s + 57.8·15-s − 77.6·16-s + 60.7·17-s + 29.1·18-s − 22.2·19-s + 47.6·20-s + 98.1·21-s − 35.5·22-s + 113.·23-s − 53.7·24-s + 247.·25-s + 80.0·26-s + 27·27-s + 80.7·28-s + ⋯ |
L(s) = 1 | + 1.14·2-s + 0.577·3-s + 0.308·4-s + 1.72·5-s + 0.660·6-s + 1.76·7-s − 0.791·8-s + 0.333·9-s + 1.97·10-s − 0.301·11-s + 0.178·12-s + 0.528·13-s + 2.02·14-s + 0.996·15-s − 1.21·16-s + 0.866·17-s + 0.381·18-s − 0.268·19-s + 0.532·20-s + 1.02·21-s − 0.344·22-s + 1.03·23-s − 0.456·24-s + 1.97·25-s + 0.604·26-s + 0.192·27-s + 0.545·28-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
\(L(2)\) |
\(\approx\) |
\(8.704775337\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.704775337\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.23T + 8T^{2} \) |
| 5 | \( 1 - 19.2T + 125T^{2} \) |
| 7 | \( 1 - 32.7T + 343T^{2} \) |
| 13 | \( 1 - 24.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 60.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 22.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 294.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 261.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 287.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 551.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 165.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 89.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 769.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 39.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 348.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 147.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 911.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 227.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 742.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 18.5T + 9.12e5T^{2} \) |
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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
−8.726817104025321369256880115807, −8.175500846253522006015611315039, −7.05778034242224502813454281038, −6.10529821084208857172193688953, −5.24229631580895328183691495295, −5.07546917563757594279986139989, −3.94405934685023980666428578891, −2.89050454735013302927065993078, −2.00752754304632570134895983835, −1.27738572957680418007304332358,
1.27738572957680418007304332358, 2.00752754304632570134895983835, 2.89050454735013302927065993078, 3.94405934685023980666428578891, 5.07546917563757594279986139989, 5.24229631580895328183691495295, 6.10529821084208857172193688953, 7.05778034242224502813454281038, 8.175500846253522006015611315039, 8.726817104025321369256880115807