| L(s) = 1 | + (−1 + 2i)5-s + (−3 − 3i)7-s − 2i·11-s + (3 + 3i)13-s + (−1 + i)17-s + 4·19-s + (1 − i)23-s + (−3 − 4i)25-s + 10i·31-s + (9 − 3i)35-s + (−1 + i)37-s + 10·41-s + (5 − 5i)43-s + (3 + 3i)47-s + 11i·49-s + ⋯ |
| L(s) = 1 | + (−0.447 + 0.894i)5-s + (−1.13 − 1.13i)7-s − 0.603i·11-s + (0.832 + 0.832i)13-s + (−0.242 + 0.242i)17-s + 0.917·19-s + (0.208 − 0.208i)23-s + (−0.600 − 0.800i)25-s + 1.79i·31-s + (1.52 − 0.507i)35-s + (−0.164 + 0.164i)37-s + 1.56·41-s + (0.762 − 0.762i)43-s + (0.437 + 0.437i)47-s + 1.57i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.287618625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.287618625\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| good | 7 | \( 1 + (3 + 3i)T + 7iT^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + (-3 - 3i)T + 13iT^{2} \) |
| 17 | \( 1 + (1 - i)T - 17iT^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (-1 + i)T - 23iT^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + (-5 + 5i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3 - 3i)T + 47iT^{2} \) |
| 53 | \( 1 + (-5 - 5i)T + 53iT^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + (1 + i)T + 67iT^{2} \) |
| 71 | \( 1 + 2iT - 71T^{2} \) |
| 73 | \( 1 + (-1 - i)T + 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (5 - 5i)T - 83iT^{2} \) |
| 89 | \( 1 - 16iT - 89T^{2} \) |
| 97 | \( 1 + (3 - 3i)T - 97iT^{2} \) |
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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
−9.634862609776960870229956953185, −8.877231507493305594451770038426, −7.85831531453200871147971389693, −6.92911308460874955430353822287, −6.65867682893380635503472243121, −5.64856297424818726926114204127, −4.13865244319710575234868419509, −3.63043030763499192110220570353, −2.75114167394878521333379041703, −0.943667154249057500058320965285,
0.70681604293710395794722326455, 2.34531910175480301220125734268, 3.38279613115516373263123655981, 4.33096906362953271837171940819, 5.54327228719001466219278672796, 5.87893304226517600485372291989, 7.10455879571962736930031763066, 7.943813942401990882893960677230, 8.768206324466987590914411901827, 9.419290581031736656185909176665
