L(s) = 1 | + (1.37 − 0.336i)2-s + (1.75 − 1.01i)3-s + (1.77 − 0.925i)4-s + (−0.5 + 0.866i)5-s + (2.06 − 1.98i)6-s + (−1.63 + 2.07i)7-s + (2.12 − 1.86i)8-s + (0.552 − 0.957i)9-s + (−0.394 + 1.35i)10-s + (−0.572 − 0.991i)11-s + (2.17 − 3.42i)12-s − 0.714·13-s + (−1.55 + 3.40i)14-s + 2.02i·15-s + (2.28 − 3.28i)16-s + (−1.98 + 1.14i)17-s + ⋯ |
L(s) = 1 | + (0.971 − 0.238i)2-s + (1.01 − 0.584i)3-s + (0.886 − 0.462i)4-s + (−0.223 + 0.387i)5-s + (0.844 − 0.809i)6-s + (−0.619 + 0.785i)7-s + (0.750 − 0.660i)8-s + (0.184 − 0.319i)9-s + (−0.124 + 0.429i)10-s + (−0.172 − 0.299i)11-s + (0.627 − 0.987i)12-s − 0.198·13-s + (−0.414 + 0.910i)14-s + 0.523i·15-s + (0.571 − 0.820i)16-s + (−0.480 + 0.277i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.54442 - 0.782298i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.54442 - 0.782298i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 + 0.336i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (1.63 - 2.07i)T \) |
good | 3 | \( 1 + (-1.75 + 1.01i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (0.572 + 0.991i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 0.714T + 13T^{2} \) |
| 17 | \( 1 + (1.98 - 1.14i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.36 + 1.94i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.00 + 2.31i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.54iT - 29T^{2} \) |
| 31 | \( 1 + (0.590 + 1.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.72 - 3.30i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.9iT - 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + (-2.60 + 4.51i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.25 + 1.30i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-10.5 + 6.07i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.13 - 12.3i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.08 - 7.08i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 15.8iT - 71T^{2} \) |
| 73 | \( 1 + (4.71 - 2.72i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.20 + 4.73i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.78iT - 83T^{2} \) |
| 89 | \( 1 + (11.0 + 6.38i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.18iT - 97T^{2} \) |
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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
−11.99957678374107560090405566739, −11.06828115688559334733772788456, −9.975040323007251759318621460094, −8.778110666528290233851400049873, −7.80368865850929674752492275817, −6.70556376547476429967277219605, −5.84291278987608569795615618383, −4.27162375058057279323970547400, −2.94195860287825898168219460487, −2.25863658596009434699407873201,
2.47500164934060602156504032633, 3.79562641495840391406518863417, 4.32285453708180814588510066801, 5.81882552991286785910782197756, 7.08557300810220557917067499898, 7.964840777303883597882855476573, 9.050780527909616074189059030546, 10.06656521116590654082188202792, 11.02344102269091129394323763299, 12.30217524390926230386748067185