| L(s) = 1 | − 0.630·2-s + 3-s − 1.60·4-s + 2.80·5-s − 0.630·6-s − 7-s + 2.27·8-s + 9-s − 1.77·10-s − 3.03·11-s − 1.60·12-s − 3.76·13-s + 0.630·14-s + 2.80·15-s + 1.77·16-s − 9.66e − 5·17-s − 0.630·18-s + 0.0432·19-s − 4.50·20-s − 21-s + 1.91·22-s + 7.26·23-s + 2.27·24-s + 2.89·25-s + 2.37·26-s + 27-s + 1.60·28-s + ⋯ |
| L(s) = 1 | − 0.446·2-s + 0.577·3-s − 0.801·4-s + 1.25·5-s − 0.257·6-s − 0.377·7-s + 0.803·8-s + 0.333·9-s − 0.560·10-s − 0.914·11-s − 0.462·12-s − 1.04·13-s + 0.168·14-s + 0.725·15-s + 0.442·16-s − 2.34e − 5·17-s − 0.148·18-s + 0.00992·19-s − 1.00·20-s − 0.218·21-s + 0.407·22-s + 1.51·23-s + 0.463·24-s + 0.578·25-s + 0.465·26-s + 0.192·27-s + 0.302·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 + 0.630T + 2T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 11 | \( 1 + 3.03T + 11T^{2} \) |
| 13 | \( 1 + 3.76T + 13T^{2} \) |
| 17 | \( 1 + 9.66e-5T + 17T^{2} \) |
| 19 | \( 1 - 0.0432T + 19T^{2} \) |
| 23 | \( 1 - 7.26T + 23T^{2} \) |
| 29 | \( 1 + 7.05T + 29T^{2} \) |
| 31 | \( 1 + 2.06T + 31T^{2} \) |
| 37 | \( 1 + 0.331T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 7.61T + 47T^{2} \) |
| 53 | \( 1 + 1.65T + 53T^{2} \) |
| 59 | \( 1 + 3.16T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 6.58T + 71T^{2} \) |
| 73 | \( 1 - 3.12T + 73T^{2} \) |
| 79 | \( 1 + 1.30T + 79T^{2} \) |
| 83 | \( 1 - 2.33T + 83T^{2} \) |
| 89 | \( 1 - 0.193T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−7.47667515932580451761273021404, −7.10089673866432319833385593467, −5.98442410063378532674236674060, −5.31072443911394121354435205441, −4.84266566609700896686629173268, −3.86051264294900940255906367693, −2.88792382411467106157223889352, −2.25130027764738281245299320635, −1.28883172146353784731366000371, 0,
1.28883172146353784731366000371, 2.25130027764738281245299320635, 2.88792382411467106157223889352, 3.86051264294900940255906367693, 4.84266566609700896686629173268, 5.31072443911394121354435205441, 5.98442410063378532674236674060, 7.10089673866432319833385593467, 7.47667515932580451761273021404
