L(s) = 1 | + 1.73·3-s + (1.73 − 2i)7-s + 2.99·9-s + (4.5 + 2.59i)11-s + (2.59 + 1.5i)17-s + (−1.5 + 0.866i)19-s + (2.99 − 3.46i)21-s + (−2.59 − 4.5i)23-s + 5.19·27-s + (−1.5 − 0.866i)31-s + (7.79 + 4.5i)33-s + (−6.06 + 3.5i)37-s + 6·41-s + 4i·43-s + (−2.59 + 1.5i)47-s + ⋯ |
L(s) = 1 | + 1.00·3-s + (0.654 − 0.755i)7-s + 0.999·9-s + (1.35 + 0.783i)11-s + (0.630 + 0.363i)17-s + (−0.344 + 0.198i)19-s + (0.654 − 0.755i)21-s + (−0.541 − 0.938i)23-s + 1.00·27-s + (−0.269 − 0.155i)31-s + (1.35 + 0.783i)33-s + (−0.996 + 0.575i)37-s + 0.937·41-s + 0.609i·43-s + (−0.378 + 0.218i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.111949043\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.111949043\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
good | 11 | \( 1 + (-4.5 - 2.59i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-2.59 - 1.5i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 + 4.5i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.06 - 3.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (2.59 - 1.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.59 + 4.5i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (10.5 - 6.06i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.33 - 2.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (-6.06 + 10.5i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + (4.5 + 7.79i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.92T + 97T^{2} \) |
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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
−8.977540157762406395696914456462, −8.348488756940266449607824689052, −7.57891493484544601645335788004, −6.96185828888425626033432062650, −6.11590173141358113435337909537, −4.67949374212584410767829872163, −4.17849430268801223419486782381, −3.37294636723242062408804669758, −2.04581514374558732256405605944, −1.26257477679841488074544912869,
1.28295683676576889501872977007, 2.21206724306692298601813684845, 3.33589164147039848152608507416, 3.99285191515913570089755753887, 5.08734011940768840149435933541, 5.97195491624406278477028424106, 6.88659130775703887294448424387, 7.77363497819103647830616856825, 8.390500013846390585104252692873, 9.187897803976598671407402709190