| L(s) = 1 | − 21.1·5-s + (−13.5 + 12.5i)7-s − 24.8i·11-s − 60.6i·13-s − 117.·17-s − 104. i·19-s + 20.1i·23-s + 320.·25-s − 28.0i·29-s − 250. i·31-s + (286. − 265. i)35-s + 18.2·37-s + 306.·41-s − 149.·43-s + 217.·47-s + ⋯ |
| L(s) = 1 | − 1.88·5-s + (−0.733 + 0.679i)7-s − 0.681i·11-s − 1.29i·13-s − 1.67·17-s − 1.26i·19-s + 0.183i·23-s + 2.56·25-s − 0.179i·29-s − 1.44i·31-s + (1.38 − 1.28i)35-s + 0.0809·37-s + 1.16·41-s − 0.528·43-s + 0.674·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3740642589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3740642589\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (13.5 - 12.5i)T \) |
| good | 5 | \( 1 + 21.1T + 125T^{2} \) |
| 11 | \( 1 + 24.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 60.6iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 104. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 20.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 28.0iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 250. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 18.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 306.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 149.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 217.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 116. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 76.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 570. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 67.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 796. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 710. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 80.1T + 4.93e5T^{2} \) |
| 83 | \( 1 + 115.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 513. iT - 9.12e5T^{2} \) |
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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
−8.192866848288752181254413070576, −7.51675644106227566308743409377, −6.71270080866761854046900586021, −5.89244271378790491471558947196, −4.82967652635998326761230053559, −4.05948496961823479421136176896, −3.18516883835447171354303872669, −2.54858577236242091483470694734, −0.48157038717277669065715265215, −0.16976374735942981162168226313,
1.19995794507054704561893343150, 2.59729294250080737036406398439, 3.83966852242896550757352148327, 4.08141474696177825578750197020, 4.85001495781838415930846396402, 6.37949743077628466488899251513, 7.02140706984385082987557220388, 7.41036425347189837641225888183, 8.428654387916850167562826154129, 8.954756565602882168717495663042
