| L(s) = 1 | − 2.23·5-s + 12.1i·7-s + 5.52i·11-s − 20.4·13-s + 1.05·17-s + 12i·19-s − 31.5i·23-s + 5.00·25-s − 44.8·29-s + 27.4i·31-s − 27.2i·35-s − 23.5·37-s − 8.69·41-s + 26.6i·43-s + 44.5i·47-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.74i·7-s + 0.502i·11-s − 1.57·13-s + 0.0621·17-s + 0.631i·19-s − 1.37i·23-s + 0.200·25-s − 1.54·29-s + 0.884i·31-s − 0.778i·35-s − 0.635·37-s − 0.212·41-s + 0.619i·43-s + 0.947i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1933539791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1933539791\) |
| \(L(2)\) |
|
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 | \( 1 + 2.23T \) |
| good | 7 | \( 1 - 12.1iT - 49T^{2} \) |
| 11 | \( 1 - 5.52iT - 121T^{2} \) |
| 13 | \( 1 + 20.4T + 169T^{2} \) |
| 17 | \( 1 - 1.05T + 289T^{2} \) |
| 19 | \( 1 - 12iT - 361T^{2} \) |
| 23 | \( 1 + 31.5iT - 529T^{2} \) |
| 29 | \( 1 + 44.8T + 841T^{2} \) |
| 31 | \( 1 - 27.4iT - 961T^{2} \) |
| 37 | \( 1 + 23.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 8.69T + 1.68e3T^{2} \) |
| 43 | \( 1 - 26.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 44.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 0.695T + 2.80e3T^{2} \) |
| 59 | \( 1 + 94.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 33.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 82.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 122. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 132.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 134. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 19.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 30T + 7.92e3T^{2} \) |
| 97 | \( 1 - 12.3T + 9.40e3T^{2} \) |
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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
−8.385149198320983410948491997804, −7.76103339247069631552486763314, −6.90457082374542512703490444704, −6.09922371775244839076416165084, −5.17020793827745118575324840159, −4.74594286892136054846177623555, −3.49413803290195172382841587122, −2.53024355242762390826059544053, −1.89728733934897548838526232046, −0.05532850939668133257867376504,
0.803606050887960966620139997088, 2.11459734154683915736418630298, 3.40090868965120492212228267721, 3.94799710009694289020138540346, 4.83574969735825827300035103029, 5.60022282226047836491450202577, 6.83544389017754698631301792455, 7.40365203956779492509965346134, 7.67750196632857743791935468544, 8.779136497457735048428607300756
