| L(s) = 1 | + i·2-s − 4-s + (1.67 + 1.48i)5-s + 2.96i·7-s − i·8-s + (−1.48 + 1.67i)10-s + 3.35·11-s + 4.96i·13-s − 2.96·14-s + 16-s − 1.35i·17-s + 4.96·19-s + (−1.67 − 1.48i)20-s + 3.35i·22-s − i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (0.749 + 0.662i)5-s + 1.11i·7-s − 0.353i·8-s + (−0.468 + 0.529i)10-s + 1.01·11-s + 1.37i·13-s − 0.791·14-s + 0.250·16-s − 0.327i·17-s + 1.13·19-s + (−0.374 − 0.331i)20-s + 0.714i·22-s − 0.208i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.051332858\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.051332858\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.67 - 1.48i)T \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - 2.96iT - 7T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 13 | \( 1 - 4.96iT - 13T^{2} \) |
| 17 | \( 1 + 1.35iT - 17T^{2} \) |
| 19 | \( 1 - 4.96T + 19T^{2} \) |
| 29 | \( 1 - 7.73T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 7.61iT - 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 3.22iT - 47T^{2} \) |
| 53 | \( 1 + 6.96iT - 53T^{2} \) |
| 59 | \( 1 - 1.22T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 7.61iT - 67T^{2} \) |
| 71 | \( 1 - 2.18T + 71T^{2} \) |
| 73 | \( 1 - 9.92iT - 73T^{2} \) |
| 79 | \( 1 - 4.12T + 79T^{2} \) |
| 83 | \( 1 + 6.38iT - 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 - 12.8iT - 97T^{2} \) |
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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
−9.263573780730106816970654926009, −8.875745763161599392389589688002, −7.76613881978788397618962760181, −6.76730226766625023860063225929, −6.46240879005604584424606293817, −5.59061975299435529725024476182, −4.82520197243561218717094726073, −3.68029564896551410281838572000, −2.60923961859417243008665541924, −1.54210725477566430726141805577,
0.817900763930529284625111189864, 1.48594427881531650920014808933, 2.93448425114306127479253478105, 3.77696183356296670717333385074, 4.70105828538587025191906836176, 5.46522609078888871220164822218, 6.37188428125783982365242314806, 7.35625112108796084755527317676, 8.209846574035499903911367527332, 8.968780127332559145071553952715
