L(s) = 1 | + 2-s + 1.56·3-s + 4-s + 5-s + 1.56·6-s − 1.82·7-s + 8-s − 0.548·9-s + 10-s − 1.63·11-s + 1.56·12-s − 5.52·13-s − 1.82·14-s + 1.56·15-s + 16-s − 4.66·17-s − 0.548·18-s − 1.04·19-s + 20-s − 2.85·21-s − 1.63·22-s − 4.03·23-s + 1.56·24-s + 25-s − 5.52·26-s − 5.55·27-s − 1.82·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.904·3-s + 0.5·4-s + 0.447·5-s + 0.639·6-s − 0.689·7-s + 0.353·8-s − 0.182·9-s + 0.316·10-s − 0.492·11-s + 0.452·12-s − 1.53·13-s − 0.487·14-s + 0.404·15-s + 0.250·16-s − 1.13·17-s − 0.129·18-s − 0.240·19-s + 0.223·20-s − 0.623·21-s − 0.348·22-s − 0.842·23-s + 0.319·24-s + 0.200·25-s − 1.08·26-s − 1.06·27-s − 0.344·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 + 1.63T + 11T^{2} \) |
| 13 | \( 1 + 5.52T + 13T^{2} \) |
| 17 | \( 1 + 4.66T + 17T^{2} \) |
| 19 | \( 1 + 1.04T + 19T^{2} \) |
| 23 | \( 1 + 4.03T + 23T^{2} \) |
| 29 | \( 1 + 3.68T + 29T^{2} \) |
| 31 | \( 1 - 3.02T + 31T^{2} \) |
| 37 | \( 1 - 2.51T + 37T^{2} \) |
| 41 | \( 1 - 4.04T + 41T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 - 4.01T + 53T^{2} \) |
| 59 | \( 1 + 1.28T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 + 9.29T + 67T^{2} \) |
| 71 | \( 1 + 2.09T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 0.431T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 - 9.78T + 97T^{2} \) |
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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
−7.936190580109977158847213722864, −7.40667995392279962404939164257, −6.50321183092528061203561067838, −5.88850706252702553505599303838, −4.98273727660931861176344175120, −4.24503190625454040111739621983, −3.26473566722828181532576868142, −2.51933275589975312491599921745, −2.06635126548649286542377262228, 0,
2.06635126548649286542377262228, 2.51933275589975312491599921745, 3.26473566722828181532576868142, 4.24503190625454040111739621983, 4.98273727660931861176344175120, 5.88850706252702553505599303838, 6.50321183092528061203561067838, 7.40667995392279962404939164257, 7.936190580109977158847213722864