L(s) = 1 | − 2-s + 4-s + 5-s + 7-s − 8-s − 10-s + 11-s − 3.06·13-s − 14-s + 16-s + 5.06·17-s − 3.38·19-s + 20-s − 22-s − 8.58·23-s + 25-s + 3.06·26-s + 28-s − 4.12·29-s + 1.38·31-s − 32-s − 5.06·34-s + 35-s + 1.06·37-s + 3.38·38-s − 40-s + 0.610·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s + 0.301·11-s − 0.849·13-s − 0.267·14-s + 0.250·16-s + 1.22·17-s − 0.777·19-s + 0.223·20-s − 0.213·22-s − 1.78·23-s + 0.200·25-s + 0.600·26-s + 0.188·28-s − 0.766·29-s + 0.249·31-s − 0.176·32-s − 0.868·34-s + 0.169·35-s + 0.174·37-s + 0.549·38-s − 0.158·40-s + 0.0953·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.463968477\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.463968477\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 3.06T + 13T^{2} \) |
| 17 | \( 1 - 5.06T + 17T^{2} \) |
| 19 | \( 1 + 3.38T + 19T^{2} \) |
| 23 | \( 1 + 8.58T + 23T^{2} \) |
| 29 | \( 1 + 4.12T + 29T^{2} \) |
| 31 | \( 1 - 1.38T + 31T^{2} \) |
| 37 | \( 1 - 1.06T + 37T^{2} \) |
| 41 | \( 1 - 0.610T + 41T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 9.51T + 53T^{2} \) |
| 59 | \( 1 - 5.38T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 15.9T + 67T^{2} \) |
| 71 | \( 1 - 14.2T + 71T^{2} \) |
| 73 | \( 1 + 3.38T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 2.73T + 97T^{2} \) |
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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
−7.989182028975774977709344064574, −7.40606822991953746613073828597, −6.65736375018455513234698414203, −5.86528822037508913396607550244, −5.34284418741521565314573579884, −4.31692720720859001491031399357, −3.53070941439505275352262703978, −2.37688289244150345560057126046, −1.85497288173185631919011707290, −0.68089779953372271872870901000,
0.68089779953372271872870901000, 1.85497288173185631919011707290, 2.37688289244150345560057126046, 3.53070941439505275352262703978, 4.31692720720859001491031399357, 5.34284418741521565314573579884, 5.86528822037508913396607550244, 6.65736375018455513234698414203, 7.40606822991953746613073828597, 7.989182028975774977709344064574