L(s) = 1 | + 2-s − 3.06·3-s + 4-s + 5-s − 3.06·6-s + 0.302·7-s + 8-s + 6.36·9-s + 10-s + 0.393·11-s − 3.06·12-s − 2.37·13-s + 0.302·14-s − 3.06·15-s + 16-s − 0.379·17-s + 6.36·18-s + 0.653·19-s + 20-s − 0.926·21-s + 0.393·22-s + 0.240·23-s − 3.06·24-s + 25-s − 2.37·26-s − 10.3·27-s + 0.302·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.76·3-s + 0.5·4-s + 0.447·5-s − 1.24·6-s + 0.114·7-s + 0.353·8-s + 2.12·9-s + 0.316·10-s + 0.118·11-s − 0.883·12-s − 0.659·13-s + 0.0808·14-s − 0.790·15-s + 0.250·16-s − 0.0921·17-s + 1.50·18-s + 0.149·19-s + 0.223·20-s − 0.202·21-s + 0.0839·22-s + 0.0500·23-s − 0.624·24-s + 0.200·25-s − 0.466·26-s − 1.98·27-s + 0.0571·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 3.06T + 3T^{2} \) |
| 7 | \( 1 - 0.302T + 7T^{2} \) |
| 11 | \( 1 - 0.393T + 11T^{2} \) |
| 13 | \( 1 + 2.37T + 13T^{2} \) |
| 17 | \( 1 + 0.379T + 17T^{2} \) |
| 19 | \( 1 - 0.653T + 19T^{2} \) |
| 23 | \( 1 - 0.240T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 - 1.32T + 31T^{2} \) |
| 37 | \( 1 - 8.99T + 37T^{2} \) |
| 41 | \( 1 + 8.36T + 41T^{2} \) |
| 43 | \( 1 + 0.545T + 43T^{2} \) |
| 47 | \( 1 + 4.02T + 47T^{2} \) |
| 53 | \( 1 + 5.16T + 53T^{2} \) |
| 59 | \( 1 + 5.74T + 59T^{2} \) |
| 61 | \( 1 - 2.10T + 61T^{2} \) |
| 67 | \( 1 - 16.1T + 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 6.29T + 79T^{2} \) |
| 83 | \( 1 - 0.717T + 83T^{2} \) |
| 89 | \( 1 + 9.68T + 89T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−7.36966056135226675345390005920, −6.72708996718194146236896218391, −6.18025545827500021183713433082, −5.49668018342093502456474099488, −5.01897461022452343781374271396, −4.38308639535108385855533121294, −3.44821411026820416890363785563, −2.19123661290156541978071275975, −1.29936114242102690669681516635, 0,
1.29936114242102690669681516635, 2.19123661290156541978071275975, 3.44821411026820416890363785563, 4.38308639535108385855533121294, 5.01897461022452343781374271396, 5.49668018342093502456474099488, 6.18025545827500021183713433082, 6.72708996718194146236896218391, 7.36966056135226675345390005920