L(s) = 1 | + (−1.5 − 2.59i)3-s + 5-s + (1.5 − 2.59i)7-s + (−3 + 5.19i)9-s + (1.5 + 2.59i)11-s + (1 − 3.46i)13-s + (−1.5 − 2.59i)15-s + (3.5 − 6.06i)17-s + (0.5 − 0.866i)19-s − 9·21-s + (−3.5 − 6.06i)23-s + 25-s + 9·27-s + (2.5 + 4.33i)29-s + 4·31-s + ⋯ |
L(s) = 1 | + (−0.866 − 1.49i)3-s + 0.447·5-s + (0.566 − 0.981i)7-s + (−1 + 1.73i)9-s + (0.452 + 0.783i)11-s + (0.277 − 0.960i)13-s + (−0.387 − 0.670i)15-s + (0.848 − 1.47i)17-s + (0.114 − 0.198i)19-s − 1.96·21-s + (−0.729 − 1.26i)23-s + 0.200·25-s + 1.73·27-s + (0.464 + 0.804i)29-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.283551096\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.283551096\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + (-1 + 3.46i)T \) |
good | 3 | \( 1 + (1.5 + 2.59i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.5 + 6.06i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.5 - 4.33i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.5 + 6.06i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.5 - 7.79i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (-2.5 + 4.33i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.5 - 2.59i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (3.5 + 6.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.5 + 9.52i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.851348328477962280274863918862, −8.395261910831623922141585940882, −7.70735351090577960309587627669, −6.98326482577137717952379021943, −6.41229212214052984800744432707, −5.33954242744985514883525626314, −4.62246804261990228396112262655, −2.89283832088833558021356842376, −1.57669554969915230220887290634, −0.70186031929410527275799333908,
1.66806304248655314634635255862, 3.37607624437404819989903804811, 4.17723899074870107867192432398, 5.19405892187006560771551621776, 5.89046720458883715911743777598, 6.35029149267896595404720225213, 8.130875103746570945378623723317, 8.794326596805942590056153439152, 9.651221308719995037582692903291, 10.15516183214452789590971172836