L(s) = 1 | + (−0.707 + 0.707i)3-s + (2.22 + 0.224i)5-s + (0.317 + 0.317i)7-s − 1.00i·9-s + 1.41i·11-s + (2.44 + 2.44i)13-s + (−1.73 + 1.41i)15-s + (−0.449 + 0.449i)17-s + 6.29·19-s − 0.449·21-s + (−4.87 + 4.87i)23-s + (4.89 + i)25-s + (0.707 + 0.707i)27-s − 5.34i·29-s − 8.48i·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.994 + 0.100i)5-s + (0.120 + 0.120i)7-s − 0.333i·9-s + 0.426i·11-s + (0.679 + 0.679i)13-s + (−0.447 + 0.365i)15-s + (−0.109 + 0.109i)17-s + 1.44·19-s − 0.0980·21-s + (−1.01 + 1.01i)23-s + (0.979 + 0.200i)25-s + (0.136 + 0.136i)27-s − 0.993i·29-s − 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22899 + 0.417356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22899 + 0.417356i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (-2.22 - 0.224i)T \) |
good | 7 | \( 1 + (-0.317 - 0.317i)T + 7iT^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + (-2.44 - 2.44i)T + 13iT^{2} \) |
| 17 | \( 1 + (0.449 - 0.449i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.29T + 19T^{2} \) |
| 23 | \( 1 + (4.87 - 4.87i)T - 23iT^{2} \) |
| 29 | \( 1 + 5.34iT - 29T^{2} \) |
| 31 | \( 1 + 8.48iT - 31T^{2} \) |
| 37 | \( 1 + (6.44 - 6.44i)T - 37iT^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + (-6.29 + 6.29i)T - 43iT^{2} \) |
| 47 | \( 1 + (8.34 + 8.34i)T + 47iT^{2} \) |
| 53 | \( 1 + (2 + 2i)T + 53iT^{2} \) |
| 59 | \( 1 - 9.89T + 59T^{2} \) |
| 61 | \( 1 + 6.89T + 61T^{2} \) |
| 67 | \( 1 + (-0.635 - 0.635i)T + 67iT^{2} \) |
| 71 | \( 1 - 1.27iT - 71T^{2} \) |
| 73 | \( 1 + (-3 - 3i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.09T + 79T^{2} \) |
| 83 | \( 1 + (1.41 - 1.41i)T - 83iT^{2} \) |
| 89 | \( 1 + 2iT - 89T^{2} \) |
| 97 | \( 1 + (-3 + 3i)T - 97iT^{2} \) |
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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
−11.95301964627212477744839914842, −11.39811336785794407205559839296, −10.02922072244405860125069596159, −9.687219047514281974711147926930, −8.472024460358868642056796997467, −7.04992601705968311591090175986, −5.98337766585449843441420157296, −5.12146012154965946890971268222, −3.67990078789532933993168205903, −1.85924119390877560639412830447,
1.37484720226801905958451630843, 3.12002058493849750662825070748, 5.00692857403998576349037879401, 5.88148364475025713589300536145, 6.84423069496962298958575007876, 8.108975387994200193022926307656, 9.120464500720444259758478829847, 10.28737005839941527815425014777, 10.94797876536633699872132365300, 12.16751772038577914333851423431