L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (2.59 − 0.5i)7-s + 0.999i·8-s + (−1.5 + 2.59i)9-s + (1 + 1.73i)11-s + (2.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (6.06 − 3.5i)17-s + (−2.59 + 1.5i)18-s + 1.99i·22-s + (−2.59 − 1.5i)23-s + (1.73 + 2i)28-s − 6·29-s + (3.5 + 6.06i)31-s + (−0.866 + 0.499i)32-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.981 − 0.188i)7-s + 0.353i·8-s + (−0.5 + 0.866i)9-s + (0.301 + 0.522i)11-s + (0.668 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (1.47 − 0.848i)17-s + (−0.612 + 0.353i)18-s + 0.426i·22-s + (−0.541 − 0.312i)23-s + (0.327 + 0.377i)28-s − 1.11·29-s + (0.628 + 1.08i)31-s + (−0.153 + 0.0883i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.82188 + 0.810383i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82188 + 0.810383i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.59 + 0.5i)T \) |
good | 3 | \( 1 + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + (-6.06 + 3.5i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.46 + 2i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 7T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + (6.06 + 3.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.46 + 2i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7 + 12.1i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.3 - 6i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + T + 71T^{2} \) |
| 73 | \( 1 + (-12.1 + 7i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 14iT - 83T^{2} \) |
| 89 | \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 7iT - 97T^{2} \) |
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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
−11.75154790135493148701774041525, −10.86660626649499448407734668610, −9.872237148049605144564518610235, −8.496063414300731290279259296454, −7.77570836985041152075369572598, −6.89638452408749512590416418684, −5.42021458664854185025725445700, −4.90789011547799026148635882731, −3.51770189566314867206876180782, −1.96514011204830873215476330558,
1.44984773523033436262610638422, 3.13890896507679599036655756324, 4.16866772115452463122730823764, 5.53576678643747861766465223510, 6.15513546850204214880593903738, 7.64170337091796259960361270332, 8.553023896009665922365098759587, 9.637351545592244872028381741092, 10.63526951691927470177616735202, 11.73580952950270820306793564199