L(s) = 1 | + (1 − 1.73i)2-s + (−1.99 − 3.46i)4-s + (−2.5 − 4.33i)5-s + (2 − 3.46i)7-s − 7.99·8-s − 10·10-s + (21 − 36.3i)11-s + (−10 − 17.3i)13-s + (−3.99 − 6.92i)14-s + (−8 + 13.8i)16-s − 93·17-s + 59·19-s + (−10 + 17.3i)20-s + (−42 − 72.7i)22-s + (4.5 + 7.79i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (0.107 − 0.187i)7-s − 0.353·8-s − 0.316·10-s + (0.575 − 0.996i)11-s + (−0.213 − 0.369i)13-s + (−0.0763 − 0.132i)14-s + (−0.125 + 0.216i)16-s − 1.32·17-s + 0.712·19-s + (−0.111 + 0.193i)20-s + (−0.407 − 0.704i)22-s + (0.0407 + 0.0706i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9240685866\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9240685866\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.5 + 4.33i)T \) |
good | 7 | \( 1 + (-2 + 3.46i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-21 + 36.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (10 + 17.3i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + 93T + 4.91e3T^{2} \) |
| 19 | \( 1 - 59T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-4.5 - 7.79i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-60 + 103. i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (23.5 + 40.7i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 262T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-63 - 109. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-89 + 154. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-72 + 124. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 741T + 1.48e5T^{2} \) |
| 59 | \( 1 + (222 + 384. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (110.5 - 191. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-269 - 465. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 690T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.12e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + (332.5 - 575. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-37.5 + 64.9i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (772 - 1.33e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−9.305636941230849243106292939456, −8.703087283481579300676929845677, −7.71849758412349843201552210072, −6.57794196636827774390423265585, −5.65332578864454022220749837141, −4.65785461220159410841570971472, −3.79351773242808896595876118518, −2.75073584178580522709338437894, −1.37046784436751794002850280141, −0.21669737127033303428434861931,
1.78585113414510919051822459903, 3.07844079670064303598111443721, 4.26648962104952081694893654842, 4.93664643501540634929347995162, 6.17027099838268383320407887343, 6.96254969412827527707059194670, 7.51465468573401674226038709066, 8.716974368978870460528275916347, 9.310517363530664891794130067158, 10.35244089591395288850059947338