L(s) = 1 | + (0.5 + 0.866i)2-s + (1.72 + 0.178i)3-s + (−0.499 + 0.866i)4-s + (−0.792 + 2.09i)5-s + (0.707 + 1.58i)6-s + (1.41 − 2.23i)7-s − 0.999·8-s + (2.93 + 0.614i)9-s + (−2.20 + 0.358i)10-s + (0.184 + 0.106i)11-s + (−1.01 + 1.40i)12-s − 6.70·13-s + (2.64 + 0.106i)14-s + (−1.73 + 3.46i)15-s + (−0.5 − 0.866i)16-s + (2.73 + 1.58i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.994 + 0.102i)3-s + (−0.249 + 0.433i)4-s + (−0.354 + 0.935i)5-s + (0.288 + 0.645i)6-s + (0.534 − 0.845i)7-s − 0.353·8-s + (0.978 + 0.204i)9-s + (−0.697 + 0.113i)10-s + (0.0557 + 0.0321i)11-s + (−0.293 + 0.404i)12-s − 1.85·13-s + (0.706 + 0.0285i)14-s + (−0.448 + 0.893i)15-s + (−0.125 − 0.216i)16-s + (0.664 + 0.383i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48438 + 1.00646i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48438 + 1.00646i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-1.72 - 0.178i)T \) |
| 5 | \( 1 + (0.792 - 2.09i)T \) |
| 7 | \( 1 + (-1.41 + 2.23i)T \) |
good | 11 | \( 1 + (-0.184 - 0.106i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 6.70T + 13T^{2} \) |
| 17 | \( 1 + (-2.73 - 1.58i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.23 + 2.44i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.23 + 5.60i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.02iT - 29T^{2} \) |
| 31 | \( 1 + (0.261 + 0.150i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.17 + 3.56i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 6.70T + 41T^{2} \) |
| 43 | \( 1 + 2.02iT - 43T^{2} \) |
| 47 | \( 1 + (6.71 - 3.87i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.5 + 4.33i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.45 - 4.25i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 1.73i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.61 - 2.66i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.02iT - 71T^{2} \) |
| 73 | \( 1 + (5.99 - 10.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.261 + 0.452i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 16.7iT - 83T^{2} \) |
| 89 | \( 1 + (5.28 + 9.15i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 3.50T + 97T^{2} \) |
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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
−12.75854170286984375611909822176, −11.71244546510959662883786613920, −10.37847052685082856899265149263, −9.675426792754967568691878272092, −8.166386875321812266084825792442, −7.48712839688867568200408201578, −6.82254485270724107405839338263, −4.94212636694557161751366684966, −3.87146738803180322131081789424, −2.61278320762526120591198217731,
1.75211057227583074807537261532, 3.13711725529749088328936392565, 4.57988624968612501641962232993, 5.46531634932315718546955915580, 7.50826216349596677668248789293, 8.220517474896512741513391643880, 9.476221151570135927537119212853, 9.825673752347047503957531259520, 11.83094451720145861103023110280, 12.01148321019130926822102296167