L(s) = 1 | + 1.85·2-s + (1.14 − 1.98i)3-s + 1.45·4-s + (−0.0986 + 0.170i)5-s + (2.13 − 3.69i)6-s − 1.01·8-s + (−1.13 − 1.95i)9-s + (−0.183 + 0.317i)10-s + (2.09 − 3.62i)11-s + (1.66 − 2.88i)12-s + (2.72 − 2.36i)13-s + (0.226 + 0.392i)15-s − 4.79·16-s − 0.841·17-s + (−2.10 − 3.64i)18-s + (0.675 + 1.17i)19-s + ⋯ |
L(s) = 1 | + 1.31·2-s + (0.662 − 1.14i)3-s + 0.726·4-s + (−0.0441 + 0.0764i)5-s + (0.870 − 1.50i)6-s − 0.359·8-s + (−0.377 − 0.653i)9-s + (−0.0579 + 0.100i)10-s + (0.630 − 1.09i)11-s + (0.481 − 0.833i)12-s + (0.755 − 0.655i)13-s + (0.0584 + 0.101i)15-s − 1.19·16-s − 0.204·17-s + (−0.495 − 0.858i)18-s + (0.155 + 0.268i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.76310 - 1.96802i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.76310 - 1.96802i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2.72 + 2.36i)T \) |
good | 2 | \( 1 - 1.85T + 2T^{2} \) |
| 3 | \( 1 + (-1.14 + 1.98i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.0986 - 0.170i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.09 + 3.62i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 0.841T + 17T^{2} \) |
| 19 | \( 1 + (-0.675 - 1.17i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 4.11T + 23T^{2} \) |
| 29 | \( 1 + (-4.11 - 7.13i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.640 + 1.10i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.04T + 37T^{2} \) |
| 41 | \( 1 + (-2.69 - 4.67i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.66 - 4.61i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.83 - 10.1i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.32 + 4.02i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 6.05T + 59T^{2} \) |
| 61 | \( 1 + (5.68 + 9.84i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.69 - 11.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.98 + 5.17i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.94 - 3.36i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.36 + 9.29i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.07T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + (-9.73 + 16.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.80299919888429879540466033297, −9.285364602641729560173425186695, −8.473347051660821810571517236639, −7.70398477176143873328067776235, −6.48482440125680120288174421588, −6.07070325171585040219128455524, −4.83566968034486254794606180293, −3.50589051999421539977626693469, −2.92934278500421617752703630122, −1.35379884565889016145538646438,
2.28768504223823744189439885375, 3.52683143997779211880704356583, 4.28984917111468093101783385148, 4.71266604731963780440994132550, 6.05258996623747266684443541089, 6.87880661546915129576860941141, 8.347550413156564810009952289595, 9.157619770806648842696099015142, 9.806744252244703481406170170786, 10.76589926642652659522782672458