| L(s) = 1 | + (−0.987 − 0.156i)2-s + (−1.45 + 2.86i)3-s + (0.951 + 0.309i)4-s + (−1.70 − 1.44i)5-s + (1.88 − 2.59i)6-s + (−1.35 + 0.692i)7-s + (−0.891 − 0.453i)8-s + (−4.29 − 5.91i)9-s + (1.45 + 1.69i)10-s + (−2.70 + 1.91i)11-s + (−2.27 + 2.27i)12-s + (−0.165 + 1.04i)13-s + (1.44 − 0.471i)14-s + (6.62 − 2.77i)15-s + (0.809 + 0.587i)16-s + (0.347 + 2.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 − 0.110i)2-s + (−0.841 + 1.65i)3-s + (0.475 + 0.154i)4-s + (−0.763 − 0.646i)5-s + (0.770 − 1.06i)6-s + (−0.513 + 0.261i)7-s + (−0.315 − 0.160i)8-s + (−1.43 − 1.97i)9-s + (0.461 + 0.535i)10-s + (−0.815 + 0.578i)11-s + (−0.655 + 0.655i)12-s + (−0.0459 + 0.289i)13-s + (0.387 − 0.125i)14-s + (1.70 − 0.716i)15-s + (0.202 + 0.146i)16-s + (0.0843 + 0.532i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0127760 + 0.259569i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0127760 + 0.259569i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.987 + 0.156i)T \) |
| 5 | \( 1 + (1.70 + 1.44i)T \) |
| 11 | \( 1 + (2.70 - 1.91i)T \) |
| good | 3 | \( 1 + (1.45 - 2.86i)T + (-1.76 - 2.42i)T^{2} \) |
| 7 | \( 1 + (1.35 - 0.692i)T + (4.11 - 5.66i)T^{2} \) |
| 13 | \( 1 + (0.165 - 1.04i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.347 - 2.19i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-1.91 - 5.89i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (3.05 + 3.05i)T + 23iT^{2} \) |
| 29 | \( 1 + (1.39 - 4.30i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.32 + 1.69i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.631 + 1.24i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-2.38 + 0.774i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (6.40 - 6.40i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.99 - 2.03i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-3.75 - 0.594i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (4.27 + 1.38i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.48 + 4.79i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-5.40 + 5.40i)T - 67iT^{2} \) |
| 71 | \( 1 + (7.19 + 5.23i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (5.17 + 10.1i)T + (-42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-4.69 + 3.40i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.26 + 0.200i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 - 10.5iT - 89T^{2} \) |
| 97 | \( 1 + (-1.09 + 6.92i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−14.71410848291826741210483048135, −12.56500629262138294326311806894, −11.90161077707345767244605651399, −10.77514165409797233881275184788, −9.993414346298698896824984458966, −9.137059114900302563326172367152, −7.939922695553468557112734296621, −6.08180665822481571989990773906, −4.82611734548273963238693110692, −3.58972061556710860241487735727,
0.38318860103915472564713046990, 2.74073950861437063344604357425, 5.57569533602237322477852593893, 6.80627126285717156011577504300, 7.42044251273006912548966782620, 8.355863368724691665216265525398, 10.24402089401269273699189017363, 11.31820620900982032934295749470, 11.84666540261199428734913870080, 13.10812420035464989590685999807
