| L(s) = 1 | − 1.41i·2-s − 1.89i·3-s − 2.00·4-s − 2.67·6-s + 1.68·7-s + 2.82i·8-s + 5.41·9-s + 1.04·11-s + 3.78i·12-s + 6.02i·13-s − 2.38i·14-s + 4.00·16-s − 21.6·17-s − 7.66i·18-s + (−17.3 + 7.85i)19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.630i·3-s − 0.500·4-s − 0.446·6-s + 0.240·7-s + 0.353i·8-s + 0.601·9-s + 0.0949·11-s + 0.315i·12-s + 0.463i·13-s − 0.170i·14-s + 0.250·16-s − 1.27·17-s − 0.425i·18-s + (−0.910 + 0.413i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2353831449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2353831449\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (17.3 - 7.85i)T \) |
| good | 3 | \( 1 + 1.89iT - 9T^{2} \) |
| 7 | \( 1 - 1.68T + 49T^{2} \) |
| 11 | \( 1 - 1.04T + 121T^{2} \) |
| 13 | \( 1 - 6.02iT - 169T^{2} \) |
| 17 | \( 1 + 21.6T + 289T^{2} \) |
| 23 | \( 1 + 39.8T + 529T^{2} \) |
| 29 | \( 1 - 0.278iT - 841T^{2} \) |
| 31 | \( 1 + 19.8iT - 961T^{2} \) |
| 37 | \( 1 + 68.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 3.84iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 65.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 9.61T + 2.20e3T^{2} \) |
| 53 | \( 1 - 62.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 94.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 19.4T + 3.72e3T^{2} \) |
| 67 | \( 1 + 62.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 41.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 3.25T + 5.32e3T^{2} \) |
| 79 | \( 1 - 84.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 5.58T + 6.88e3T^{2} \) |
| 89 | \( 1 + 84.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 1.89iT - 9.40e3T^{2} \) |
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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
−9.355163325465929460242064791337, −8.471640074871961122920659455790, −7.68597874648475614839251708693, −6.70667915412341365405524146618, −5.89856468890979249111903353581, −4.47993599930463241886812954927, −3.94716862491215597466403321035, −2.29539742164071454098627624422, −1.66975552623680079537277614810, −0.07104638475867478101383061092,
1.83726451849352117644003286195, 3.45528994202120891817380336027, 4.47648660688416609240450205492, 4.99741100759166384878123452833, 6.30774721088786698711835657645, 6.85273701443307100527346228257, 8.070950174351124818517745412091, 8.568812519537877124456961877989, 9.643555880144777784230902418347, 10.18824182388356933288785096473
