L(s) = 1 | + (−2.17 + 0.582i)2-s + (2.65 − 1.53i)4-s + (1.96 − 1.06i)5-s + (−0.660 − 2.56i)7-s + (−1.69 + 1.69i)8-s + (−3.65 + 3.46i)10-s + (−0.329 − 0.571i)11-s + (−2.55 − 2.55i)13-s + (2.92 + 5.18i)14-s + (−0.366 + 0.635i)16-s + (−5.43 − 1.45i)17-s + (−1.48 + 2.56i)19-s + (3.58 − 5.84i)20-s + (1.04 + 1.04i)22-s + (−0.0726 − 0.271i)23-s + ⋯ |
L(s) = 1 | + (−1.53 + 0.411i)2-s + (1.32 − 0.766i)4-s + (0.878 − 0.476i)5-s + (−0.249 − 0.968i)7-s + (−0.599 + 0.599i)8-s + (−1.15 + 1.09i)10-s + (−0.0994 − 0.172i)11-s + (−0.709 − 0.709i)13-s + (0.782 + 1.38i)14-s + (−0.0916 + 0.158i)16-s + (−1.31 − 0.353i)17-s + (−0.339 + 0.588i)19-s + (0.801 − 1.30i)20-s + (0.223 + 0.223i)22-s + (−0.0151 − 0.0565i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.402713 - 0.353861i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.402713 - 0.353861i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.96 + 1.06i)T \) |
| 7 | \( 1 + (0.660 + 2.56i)T \) |
good | 2 | \( 1 + (2.17 - 0.582i)T + (1.73 - i)T^{2} \) |
| 11 | \( 1 + (0.329 + 0.571i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.55 + 2.55i)T + 13iT^{2} \) |
| 17 | \( 1 + (5.43 + 1.45i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.48 - 2.56i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0726 + 0.271i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 5.03iT - 29T^{2} \) |
| 31 | \( 1 + (-6.53 + 3.77i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (7.79 - 2.08i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (-8.53 + 8.53i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.11 - 11.6i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-4.65 - 1.24i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.782 - 1.35i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.02 + 0.589i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.762 + 2.84i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 3.13T + 71T^{2} \) |
| 73 | \( 1 + (-0.417 + 1.55i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-6.17 - 3.56i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.14 - 2.14i)T + 83iT^{2} \) |
| 89 | \( 1 + (-2.24 + 3.88i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.33 - 3.33i)T - 97iT^{2} \) |
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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
−10.83914227517892892374551792150, −10.27949028342204337805719472322, −9.545578807278284588854928920646, −8.705733924154276177245626199957, −7.76012869718596962723835521318, −6.85337792687938768385514999158, −5.89135107588847354432529421020, −4.39138753055008835139749539634, −2.26172786246703107369498510963, −0.61013617474111483961276740199,
1.93028802725597538904958090621, 2.72592839555339262299390399234, 4.94934035586986777444494300802, 6.42677629603395637493359639268, 7.11838517853221748218601603460, 8.588914371813497006793892837754, 9.089872569206332006160257795536, 9.895963656763889029356378292268, 10.69232808949672093124108313513, 11.53095385284746206397039367600