L(s) = 1 | + 2.42·2-s − 1.56·3-s + 3.88·4-s − 5-s − 3.78·6-s + 7-s + 4.57·8-s − 0.565·9-s − 2.42·10-s − 1.28·11-s − 6.06·12-s + 0.610·13-s + 2.42·14-s + 1.56·15-s + 3.32·16-s + 0.0371·17-s − 1.37·18-s + 3.97·19-s − 3.88·20-s − 1.56·21-s − 3.11·22-s + 1.18·23-s − 7.13·24-s + 25-s + 1.48·26-s + 5.56·27-s + 3.88·28-s + ⋯ |
L(s) = 1 | + 1.71·2-s − 0.900·3-s + 1.94·4-s − 0.447·5-s − 1.54·6-s + 0.377·7-s + 1.61·8-s − 0.188·9-s − 0.767·10-s − 0.386·11-s − 1.74·12-s + 0.169·13-s + 0.648·14-s + 0.402·15-s + 0.830·16-s + 0.00899·17-s − 0.323·18-s + 0.912·19-s − 0.868·20-s − 0.340·21-s − 0.663·22-s + 0.247·23-s − 1.45·24-s + 0.200·25-s + 0.290·26-s + 1.07·27-s + 0.734·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 - T \) |
good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 3 | \( 1 + 1.56T + 3T^{2} \) |
| 11 | \( 1 + 1.28T + 11T^{2} \) |
| 13 | \( 1 - 0.610T + 13T^{2} \) |
| 17 | \( 1 - 0.0371T + 17T^{2} \) |
| 19 | \( 1 - 3.97T + 19T^{2} \) |
| 23 | \( 1 - 1.18T + 23T^{2} \) |
| 29 | \( 1 + 7.99T + 29T^{2} \) |
| 31 | \( 1 + 9.87T + 31T^{2} \) |
| 37 | \( 1 - 0.982T + 37T^{2} \) |
| 41 | \( 1 - 3.37T + 41T^{2} \) |
| 43 | \( 1 + 5.33T + 43T^{2} \) |
| 47 | \( 1 - 4.27T + 47T^{2} \) |
| 53 | \( 1 + 8.49T + 53T^{2} \) |
| 59 | \( 1 + 5.41T + 59T^{2} \) |
| 61 | \( 1 - 6.37T + 61T^{2} \) |
| 67 | \( 1 - 8.37T + 67T^{2} \) |
| 71 | \( 1 - 3.39T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 5.20T + 79T^{2} \) |
| 83 | \( 1 + 8.62T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 3.46T + 97T^{2} \) |
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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
−7.27673446934005899681966937359, −6.53093102304588843980711878703, −5.73332717883176107386474468661, −5.38338611297615979911759489950, −4.85386764162058147998003649712, −3.98828347021831585370561676145, −3.38558719463195064059941618361, −2.55685985665757261768918789015, −1.48302808860350121773592212815, 0,
1.48302808860350121773592212815, 2.55685985665757261768918789015, 3.38558719463195064059941618361, 3.98828347021831585370561676145, 4.85386764162058147998003649712, 5.38338611297615979911759489950, 5.73332717883176107386474468661, 6.53093102304588843980711878703, 7.27673446934005899681966937359