| L(s) = 1 | + (−2.22 − 0.224i)5-s − i·7-s − 4.89·11-s + 4.44i·13-s + 2i·17-s + 1.55·19-s + 2.89i·23-s + (4.89 + i)25-s + 6.89·29-s − 8.89·31-s + (−0.224 + 2.22i)35-s − 2i·37-s + 1.10·41-s + 0.898i·43-s − 8.89i·47-s + ⋯ |
| L(s) = 1 | + (−0.994 − 0.100i)5-s − 0.377i·7-s − 1.47·11-s + 1.23i·13-s + 0.485i·17-s + 0.355·19-s + 0.604i·23-s + (0.979 + 0.200i)25-s + 1.28·29-s − 1.59·31-s + (−0.0379 + 0.376i)35-s − 0.328i·37-s + 0.171·41-s + 0.137i·43-s − 1.29i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5974712898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5974712898\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.22 + 0.224i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.44iT - 13T^{2} \) |
| 17 | \( 1 - 2iT - 17T^{2} \) |
| 19 | \( 1 - 1.55T + 19T^{2} \) |
| 23 | \( 1 - 2.89iT - 23T^{2} \) |
| 29 | \( 1 - 6.89T + 29T^{2} \) |
| 31 | \( 1 + 8.89T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 1.10T + 41T^{2} \) |
| 43 | \( 1 - 0.898iT - 43T^{2} \) |
| 47 | \( 1 + 8.89iT - 47T^{2} \) |
| 53 | \( 1 - 10.8iT - 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 - 3.55T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 1.10T + 71T^{2} \) |
| 73 | \( 1 - 2.89iT - 73T^{2} \) |
| 79 | \( 1 - 6.89T + 79T^{2} \) |
| 83 | \( 1 + 2.44iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 15.7iT - 97T^{2} \) |
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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
−8.002432134992785607005525799641, −7.33057898824800462075853528349, −6.87976995765092538930736560307, −5.78613077385758404841004733236, −5.02355848024213977896686106405, −4.29250869023658004720757587532, −3.60141931835828333824927418355, −2.70094016815109404578547354667, −1.58414999565701377324099176676, −0.21482175404544898412701125562,
0.817105609953640065201371210041, 2.48271524996292013803451496424, 2.99763760540529880106677376555, 3.85827042592478622963160508211, 5.01021407039934160574169127588, 5.24091583399066390131621550159, 6.28292480726181280911809778455, 7.18903919514920674350944881304, 7.81508886386271855235692694181, 8.226267554384505922750137649956
