| L(s) = 1 | + i·2-s − 4-s + (−0.605 + 2.15i)5-s − 2i·7-s − i·8-s + (−2.15 − 0.605i)10-s − 5.21·11-s + 0.789i·13-s + 2·14-s + 16-s − 0.115i·17-s − 0.115·19-s + (0.605 − 2.15i)20-s − 5.21i·22-s + 4.42i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.270 + 0.962i)5-s − 0.755i·7-s − 0.353i·8-s + (−0.680 − 0.191i)10-s − 1.57·11-s + 0.219i·13-s + 0.534·14-s + 0.250·16-s − 0.0279i·17-s − 0.0264·19-s + (0.135 − 0.481i)20-s − 1.11i·22-s + 0.921i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8871526829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8871526829\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.605 - 2.15i)T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 5.21T + 11T^{2} \) |
| 13 | \( 1 - 0.789iT - 13T^{2} \) |
| 17 | \( 1 + 0.115iT - 17T^{2} \) |
| 19 | \( 1 + 0.115T + 19T^{2} \) |
| 23 | \( 1 - 4.42iT - 23T^{2} \) |
| 29 | \( 1 - 4.42T + 29T^{2} \) |
| 37 | \( 1 + 6.61iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.61iT - 43T^{2} \) |
| 47 | \( 1 - 0.115iT - 47T^{2} \) |
| 53 | \( 1 - 0.190iT - 53T^{2} \) |
| 59 | \( 1 + 4.19T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 5.82iT - 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 10.8iT - 73T^{2} \) |
| 79 | \( 1 + 2.30T + 79T^{2} \) |
| 83 | \( 1 + 15.1iT - 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 - 9.21iT - 97T^{2} \) |
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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
−8.531865471832843079031692193824, −7.73136743216915636827607890341, −7.35490267610349974295255113918, −6.66595005712781025852062336763, −5.74524062593431201169227213474, −5.01168118479241522512051681908, −4.00553196143611668572057910364, −3.24261147623160967759315745089, −2.17020944403939733225694140868, −0.34962173860540742348315075489,
0.939240845505328182626541334485, 2.27441150057056916856944353846, 2.93729845933376948290806106733, 4.11821739369513475766727745998, 5.01759276391249529557978497016, 5.37165064292284556433399005516, 6.42139017399056123997164946822, 7.66078413756717069705725905061, 8.372775350947612338686126169641, 8.650563007928857652662532152088
