| L(s) = 1 | − 2.08·2-s − 0.324·3-s + 2.36·4-s − 5-s + 0.677·6-s − 7-s − 0.758·8-s − 2.89·9-s + 2.08·10-s − 3.93·11-s − 0.766·12-s − 1.67·13-s + 2.08·14-s + 0.324·15-s − 3.14·16-s − 1.01·17-s + 6.04·18-s + 1.87·19-s − 2.36·20-s + 0.324·21-s + 8.22·22-s + 0.464·23-s + 0.246·24-s + 25-s + 3.48·26-s + 1.91·27-s − 2.36·28-s + ⋯ |
| L(s) = 1 | − 1.47·2-s − 0.187·3-s + 1.18·4-s − 0.447·5-s + 0.276·6-s − 0.377·7-s − 0.268·8-s − 0.964·9-s + 0.660·10-s − 1.18·11-s − 0.221·12-s − 0.463·13-s + 0.558·14-s + 0.0837·15-s − 0.785·16-s − 0.245·17-s + 1.42·18-s + 0.429·19-s − 0.528·20-s + 0.0707·21-s + 1.75·22-s + 0.0968·23-s + 0.0502·24-s + 0.200·25-s + 0.684·26-s + 0.367·27-s − 0.446·28-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 | \( 1 + 2.08T + 2T^{2} \) |
| 3 | \( 1 + 0.324T + 3T^{2} \) |
| 11 | \( 1 + 3.93T + 11T^{2} \) |
| 13 | \( 1 + 1.67T + 13T^{2} \) |
| 17 | \( 1 + 1.01T + 17T^{2} \) |
| 19 | \( 1 - 1.87T + 19T^{2} \) |
| 23 | \( 1 - 0.464T + 23T^{2} \) |
| 29 | \( 1 + 7.71T + 29T^{2} \) |
| 31 | \( 1 - 3.42T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 - 9.56T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 4.47T + 47T^{2} \) |
| 53 | \( 1 - 7.31T + 53T^{2} \) |
| 59 | \( 1 - 8.41T + 59T^{2} \) |
| 61 | \( 1 + 9.55T + 61T^{2} \) |
| 67 | \( 1 + 3.30T + 67T^{2} \) |
| 71 | \( 1 - 2.49T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 8.14T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 - 7.25T + 89T^{2} \) |
| 97 | \( 1 - 0.907T + 97T^{2} \) |
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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
−7.76301343702197632660332328579, −7.09537320721676709901515030317, −6.32142281556464147353453494567, −5.46615060381515632635008673562, −4.83529959895679473597911396678, −3.73839718815550081827464999741, −2.75608103386524603039036774996, −2.17655670296313441989068613032, −0.77403338479583891242682929440, 0,
0.77403338479583891242682929440, 2.17655670296313441989068613032, 2.75608103386524603039036774996, 3.73839718815550081827464999741, 4.83529959895679473597911396678, 5.46615060381515632635008673562, 6.32142281556464147353453494567, 7.09537320721676709901515030317, 7.76301343702197632660332328579
