L(s) = 1 | + 4.78·2-s − 3·3-s + 14.9·4-s − 3.84·5-s − 14.3·6-s + 23.4·7-s + 33.1·8-s + 9·9-s − 18.3·10-s + 11·11-s − 44.7·12-s − 80.3·13-s + 112.·14-s + 11.5·15-s + 39.4·16-s + 84.9·17-s + 43.0·18-s + 10.7·19-s − 57.3·20-s − 70.4·21-s + 52.6·22-s + 148.·23-s − 99.5·24-s − 110.·25-s − 384.·26-s − 27·27-s + 350.·28-s + ⋯ |
L(s) = 1 | + 1.69·2-s − 0.577·3-s + 1.86·4-s − 0.343·5-s − 0.977·6-s + 1.26·7-s + 1.46·8-s + 0.333·9-s − 0.581·10-s + 0.301·11-s − 1.07·12-s − 1.71·13-s + 2.14·14-s + 0.198·15-s + 0.617·16-s + 1.21·17-s + 0.564·18-s + 0.129·19-s − 0.641·20-s − 0.732·21-s + 0.510·22-s + 1.34·23-s − 0.846·24-s − 0.881·25-s − 2.90·26-s − 0.192·27-s + 2.36·28-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
\(L(2)\) |
\(\approx\) |
\(6.000907107\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.000907107\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.78T + 8T^{2} \) |
| 5 | \( 1 + 3.84T + 125T^{2} \) |
| 7 | \( 1 - 23.4T + 343T^{2} \) |
| 13 | \( 1 + 80.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 84.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 10.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 148.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 242.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 518.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 129.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 495.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 557.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 498.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 456.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 705.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 566.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 748.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 497.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 591.T + 9.12e5T^{2} \) |
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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
−8.586805624408686458315715732406, −7.56460639218085532177384561985, −7.14203302106387722945195627166, −6.14566294486444620710824246856, −5.19404441823632328133802459969, −4.90659212204329507267732601352, −4.18077240987340612542194801600, −3.09881874540743441395135778677, −2.15827698141063325454878287824, −0.923005848573767945589638088114,
0.923005848573767945589638088114, 2.15827698141063325454878287824, 3.09881874540743441395135778677, 4.18077240987340612542194801600, 4.90659212204329507267732601352, 5.19404441823632328133802459969, 6.14566294486444620710824246856, 7.14203302106387722945195627166, 7.56460639218085532177384561985, 8.586805624408686458315715732406