| L(s) = 1 | + (0.618 + 1.90i)2-s + (2.90 + 2.11i)3-s + (−3.23 + 2.35i)4-s + (−1.54 + 4.75i)5-s + (−2.22 + 6.83i)6-s + (−21.1 + 15.3i)7-s + (−6.47 − 4.70i)8-s + (−4.34 − 13.3i)9-s − 10.0·10-s + (9.61 + 35.1i)11-s − 14.3·12-s + (3.24 + 9.98i)13-s + (−42.3 − 30.7i)14-s + (−14.5 + 10.5i)15-s + (4.94 − 15.2i)16-s + (−2.54 + 7.84i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.559 + 0.406i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (−0.151 + 0.465i)6-s + (−1.14 + 0.831i)7-s + (−0.286 − 0.207i)8-s + (−0.161 − 0.495i)9-s − 0.316·10-s + (0.263 + 0.964i)11-s − 0.346·12-s + (0.0692 + 0.212i)13-s + (−0.809 − 0.587i)14-s + (−0.250 + 0.181i)15-s + (0.0772 − 0.237i)16-s + (−0.0363 + 0.111i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 - 0.396i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.300623 + 1.45537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.300623 + 1.45537i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.618 - 1.90i)T \) |
| 5 | \( 1 + (1.54 - 4.75i)T \) |
| 11 | \( 1 + (-9.61 - 35.1i)T \) |
| good | 3 | \( 1 + (-2.90 - 2.11i)T + (8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 + (21.1 - 15.3i)T + (105. - 326. i)T^{2} \) |
| 13 | \( 1 + (-3.24 - 9.98i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (2.54 - 7.84i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-84.6 - 61.5i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-111. + 81.0i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-66.6 - 205. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (48.1 - 34.9i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-318. - 231. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 252.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-24.2 - 17.6i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-96.8 - 298. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-536. + 389. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-190. + 585. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 - 4.92T + 3.00e5T^{2} \) |
| 71 | \( 1 + (180. - 554. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (55.0 - 39.9i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (358. + 1.10e3i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (406. - 1.24e3i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + 1.25e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (194. + 599. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−14.01396703210058057170251757278, −12.56227590273226894427974376774, −11.93147382876317057439842009964, −9.952472652687857946518879850908, −9.415019344620119458140364367502, −8.207255437690570550622516047725, −6.81590709050702011731425249666, −5.87877056919326057173281682598, −4.08264897084783061326879593967, −2.88739370740910284261437921202,
0.72266740358782876254670433005, 2.77647676781909928562131629360, 3.99139808089771344815047818650, 5.72511608454692232859891207545, 7.25535991518582935821657056789, 8.484995710278364348052808064720, 9.566031505922624248829629861762, 10.64859013510394169266841789949, 11.77440457246089553178668995573, 12.97139942564235966368721526387
