| L(s) = 1 | − 3-s + 2i·7-s + 9-s + 4·13-s − 2i·17-s − 4i·19-s − 2i·21-s + 4i·23-s − 27-s − 6i·29-s − 2·31-s + 8·37-s − 4·39-s + 2·41-s − 4·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.755i·7-s + 0.333·9-s + 1.10·13-s − 0.485i·17-s − 0.917i·19-s − 0.436i·21-s + 0.834i·23-s − 0.192·27-s − 1.11i·29-s − 0.359·31-s + 1.31·37-s − 0.640·39-s + 0.312·41-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 - 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.496537743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.496537743\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.201456728366160014112418276019, −8.193324009063504683815779123636, −7.50146990945668746921517980326, −6.45370586418131396744702583291, −5.96159217921078851660842185362, −5.14064930409693311470736756091, −4.29388693324467729935957418171, −3.23402412123181115786644503591, −2.18023878975580030149348279261, −0.870041281663465352660196142108,
0.790556167973658235651492584675, 1.86261957773123590830664150246, 3.40098750945713421189680905943, 4.04913214124790909642121575831, 4.97010407515004408024544188535, 5.93246984238772117476279397343, 6.49537248118881417138241689121, 7.33173395264954375094624551930, 8.150673485653566013322444174490, 8.844441710336480391069297210112
