L(s) = 1 | + 0.618·2-s − 1.61·4-s + 5-s − 7-s − 2.23·8-s + 0.618·10-s + 3.47·11-s − 6.09·13-s − 0.618·14-s + 1.85·16-s − 6.61·17-s + 0.381·19-s − 1.61·20-s + 2.14·22-s − 4.38·23-s + 25-s − 3.76·26-s + 1.61·28-s − 2.85·29-s + 3·31-s + 5.61·32-s − 4.09·34-s − 35-s − 3·37-s + 0.236·38-s − 2.23·40-s + 0.618·41-s + ⋯ |
L(s) = 1 | + 0.437·2-s − 0.809·4-s + 0.447·5-s − 0.377·7-s − 0.790·8-s + 0.195·10-s + 1.04·11-s − 1.68·13-s − 0.165·14-s + 0.463·16-s − 1.60·17-s + 0.0876·19-s − 0.361·20-s + 0.457·22-s − 0.913·23-s + 0.200·25-s − 0.738·26-s + 0.305·28-s − 0.529·29-s + 0.538·31-s + 0.993·32-s − 0.701·34-s − 0.169·35-s − 0.493·37-s + 0.0382·38-s − 0.353·40-s + 0.0965·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 13 | \( 1 + 6.09T + 13T^{2} \) |
| 17 | \( 1 + 6.61T + 17T^{2} \) |
| 19 | \( 1 - 0.381T + 19T^{2} \) |
| 23 | \( 1 + 4.38T + 23T^{2} \) |
| 29 | \( 1 + 2.85T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 - 0.618T + 41T^{2} \) |
| 43 | \( 1 + 7.76T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 6.61T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 1.14T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 8.85T + 71T^{2} \) |
| 73 | \( 1 + 2.52T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.550660287951508541321813423167, −9.044804327219182083733636154064, −8.041983683939172592327563607475, −6.82249740246230560233059689989, −6.19626267906589685771942440380, −5.02573903522813994495500054452, −4.41379801848985815164581208006, −3.31625884416302874363695905680, −2.03468494786555335800175458069, 0,
2.03468494786555335800175458069, 3.31625884416302874363695905680, 4.41379801848985815164581208006, 5.02573903522813994495500054452, 6.19626267906589685771942440380, 6.82249740246230560233059689989, 8.041983683939172592327563607475, 9.044804327219182083733636154064, 9.550660287951508541321813423167