L(s) = 1 | − 2.31·2-s − 1.68·3-s + 3.34·4-s − 5-s + 3.89·6-s − 2.49·7-s − 3.10·8-s − 0.162·9-s + 2.31·10-s − 11-s − 5.63·12-s − 5.27·13-s + 5.76·14-s + 1.68·15-s + 0.495·16-s + 2.07·17-s + 0.376·18-s − 2.21·19-s − 3.34·20-s + 4.20·21-s + 2.31·22-s − 3.45·23-s + 5.23·24-s + 25-s + 12.2·26-s + 5.32·27-s − 8.34·28-s + ⋯ |
L(s) = 1 | − 1.63·2-s − 0.972·3-s + 1.67·4-s − 0.447·5-s + 1.58·6-s − 0.942·7-s − 1.09·8-s − 0.0543·9-s + 0.731·10-s − 0.301·11-s − 1.62·12-s − 1.46·13-s + 1.54·14-s + 0.434·15-s + 0.123·16-s + 0.504·17-s + 0.0887·18-s − 0.509·19-s − 0.747·20-s + 0.916·21-s + 0.492·22-s − 0.720·23-s + 1.06·24-s + 0.200·25-s + 2.39·26-s + 1.02·27-s − 1.57·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 + 2.31T + 2T^{2} \) |
| 3 | \( 1 + 1.68T + 3T^{2} \) |
| 7 | \( 1 + 2.49T + 7T^{2} \) |
| 13 | \( 1 + 5.27T + 13T^{2} \) |
| 17 | \( 1 - 2.07T + 17T^{2} \) |
| 19 | \( 1 + 2.21T + 19T^{2} \) |
| 23 | \( 1 + 3.45T + 23T^{2} \) |
| 29 | \( 1 - 1.15T + 29T^{2} \) |
| 31 | \( 1 - 10.3T + 31T^{2} \) |
| 37 | \( 1 + 7.87T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 2.08T + 43T^{2} \) |
| 47 | \( 1 + 7.81T + 47T^{2} \) |
| 53 | \( 1 - 8.46T + 53T^{2} \) |
| 59 | \( 1 - 13.0T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 + 3.66T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 + 2.46T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−8.230213682650533060583299340312, −7.37111621097015194248933826775, −6.81423135562917588872883672419, −6.17893344038729787162618878971, −5.26915069122669909775753416122, −4.38078733953032656415667258641, −3.04946035051038380910185238011, −2.25086148532179030284347300181, −0.75267552993475557122500231472, 0,
0.75267552993475557122500231472, 2.25086148532179030284347300181, 3.04946035051038380910185238011, 4.38078733953032656415667258641, 5.26915069122669909775753416122, 6.17893344038729787162618878971, 6.81423135562917588872883672419, 7.37111621097015194248933826775, 8.230213682650533060583299340312