| L(s) = 1 | − 2.22·2-s + 3-s + 2.95·4-s − 0.700·5-s − 2.22·6-s − 7-s − 2.13·8-s + 9-s + 1.55·10-s + 2.84·11-s + 2.95·12-s − 1.64·13-s + 2.22·14-s − 0.700·15-s − 1.16·16-s + 1.41·17-s − 2.22·18-s + 3.55·19-s − 2.07·20-s − 21-s − 6.32·22-s − 9.09·23-s − 2.13·24-s − 4.50·25-s + 3.66·26-s + 27-s − 2.95·28-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 0.577·3-s + 1.47·4-s − 0.313·5-s − 0.908·6-s − 0.377·7-s − 0.753·8-s + 0.333·9-s + 0.493·10-s + 0.856·11-s + 0.853·12-s − 0.456·13-s + 0.595·14-s − 0.180·15-s − 0.291·16-s + 0.344·17-s − 0.524·18-s + 0.814·19-s − 0.463·20-s − 0.218·21-s − 1.34·22-s − 1.89·23-s − 0.435·24-s − 0.901·25-s + 0.719·26-s + 0.192·27-s − 0.558·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 2.22T + 2T^{2} \) |
| 5 | \( 1 + 0.700T + 5T^{2} \) |
| 11 | \( 1 - 2.84T + 11T^{2} \) |
| 13 | \( 1 + 1.64T + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 + 9.09T + 23T^{2} \) |
| 29 | \( 1 - 5.14T + 29T^{2} \) |
| 31 | \( 1 + 5.38T + 31T^{2} \) |
| 37 | \( 1 + 8.77T + 37T^{2} \) |
| 41 | \( 1 - 7.79T + 41T^{2} \) |
| 43 | \( 1 + 2.96T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 1.24T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 3.35T + 61T^{2} \) |
| 67 | \( 1 - 7.32T + 67T^{2} \) |
| 71 | \( 1 + 0.417T + 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 + 0.994T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.390754458991890774954688106620, −8.009455453119513932905864018349, −7.20417913710268962121242441175, −6.63279732310566741665980020481, −5.57706810824694202319422167009, −4.24364813796058789169297505869, −3.44909568710305543275899847602, −2.28126890579600344315306951189, −1.37902683278073173456070323217, 0,
1.37902683278073173456070323217, 2.28126890579600344315306951189, 3.44909568710305543275899847602, 4.24364813796058789169297505869, 5.57706810824694202319422167009, 6.63279732310566741665980020481, 7.20417913710268962121242441175, 8.009455453119513932905864018349, 8.390754458991890774954688106620
