| L(s) = 1 | − 2-s + 4-s − 1.07·7-s − 8-s + 4·11-s − 5.41·13-s + 1.07·14-s + 16-s − 17-s + 8.68·19-s − 4·22-s − 0.921·23-s + 5.41·26-s − 1.07·28-s − 4.34·29-s + 3.07·31-s − 32-s + 34-s + 8.34·37-s − 8.68·38-s + 3.26·41-s − 9.41·43-s + 4·44-s + 0.921·46-s − 2.15·47-s − 5.83·49-s − 5.41·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.407·7-s − 0.353·8-s + 1.20·11-s − 1.50·13-s + 0.288·14-s + 0.250·16-s − 0.242·17-s + 1.99·19-s − 0.852·22-s − 0.192·23-s + 1.06·26-s − 0.203·28-s − 0.805·29-s + 0.552·31-s − 0.176·32-s + 0.171·34-s + 1.37·37-s − 1.40·38-s + 0.509·41-s − 1.43·43-s + 0.603·44-s + 0.135·46-s − 0.314·47-s − 0.833·49-s − 0.751·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.282876177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.282876177\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 1.07T + 7T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 5.41T + 13T^{2} \) |
| 19 | \( 1 - 8.68T + 19T^{2} \) |
| 23 | \( 1 + 0.921T + 23T^{2} \) |
| 29 | \( 1 + 4.34T + 29T^{2} \) |
| 31 | \( 1 - 3.07T + 31T^{2} \) |
| 37 | \( 1 - 8.34T + 37T^{2} \) |
| 41 | \( 1 - 3.26T + 41T^{2} \) |
| 43 | \( 1 + 9.41T + 43T^{2} \) |
| 47 | \( 1 + 2.15T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 7.23T + 61T^{2} \) |
| 67 | \( 1 - 2.58T + 67T^{2} \) |
| 71 | \( 1 - 3.60T + 71T^{2} \) |
| 73 | \( 1 - 6.58T + 73T^{2} \) |
| 79 | \( 1 + 12.4T + 79T^{2} \) |
| 83 | \( 1 + 8.68T + 83T^{2} \) |
| 89 | \( 1 + 6.15T + 89T^{2} \) |
| 97 | \( 1 - 2.09T + 97T^{2} \) |
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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
−7.79810933037071940945728585426, −7.21422422202007300433046450619, −6.72652856513421828900271145785, −5.89296293448296572319371209996, −5.15266658381571489675100972309, −4.27611416720214870378123787608, −3.35931897595406840323215442285, −2.64139414825280884533298749406, −1.63621416373466228554020977512, −0.64296210677065753136520315436,
0.64296210677065753136520315436, 1.63621416373466228554020977512, 2.64139414825280884533298749406, 3.35931897595406840323215442285, 4.27611416720214870378123787608, 5.15266658381571489675100972309, 5.89296293448296572319371209996, 6.72652856513421828900271145785, 7.21422422202007300433046450619, 7.79810933037071940945728585426
