| L(s) = 1 | + 1.53i·5-s − 7-s + 2.28i·11-s + 7.10i·13-s + 6.81·17-s + 1.60i·19-s − 1.16·23-s + 2.64·25-s + 4.07i·29-s + 7.90·31-s − 1.53i·35-s − 7.04i·37-s + 10.1·41-s − 0.344i·43-s − 3.10·47-s + ⋯ |
| L(s) = 1 | + 0.685i·5-s − 0.377·7-s + 0.688i·11-s + 1.97i·13-s + 1.65·17-s + 0.368i·19-s − 0.243·23-s + 0.529·25-s + 0.755i·29-s + 1.41·31-s − 0.259i·35-s − 1.15i·37-s + 1.58·41-s − 0.0525i·43-s − 0.453·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.905035520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.905035520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 1.53iT - 5T^{2} \) |
| 11 | \( 1 - 2.28iT - 11T^{2} \) |
| 13 | \( 1 - 7.10iT - 13T^{2} \) |
| 17 | \( 1 - 6.81T + 17T^{2} \) |
| 19 | \( 1 - 1.60iT - 19T^{2} \) |
| 23 | \( 1 + 1.16T + 23T^{2} \) |
| 29 | \( 1 - 4.07iT - 29T^{2} \) |
| 31 | \( 1 - 7.90T + 31T^{2} \) |
| 37 | \( 1 + 7.04iT - 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 0.344iT - 43T^{2} \) |
| 47 | \( 1 + 3.10T + 47T^{2} \) |
| 53 | \( 1 + 5.66iT - 53T^{2} \) |
| 59 | \( 1 - 1.38iT - 59T^{2} \) |
| 61 | \( 1 - 5.50iT - 61T^{2} \) |
| 67 | \( 1 + 8.35iT - 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 5.95T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 - 14.7iT - 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 2.60T + 97T^{2} \) |
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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
−8.212108408497359925675264322053, −7.42849975157135729735679075697, −6.90135245890128447960318255339, −6.31909918841194820831097461998, −5.52532141021502877005758902326, −4.55492420576931800412628938445, −3.91171453173403582186496127861, −3.04945051260353620571776760535, −2.19778226827309367266473551658, −1.21730649075790780814725802006,
0.57928690348246487872520001031, 1.17105536473991576840618359837, 2.88958921084509193326643488479, 3.08848959020706805973234391881, 4.23913367214257473386102638918, 5.07679448738036713728907689024, 5.75049273614323031322197413146, 6.17841676476322230704528905405, 7.30476964401538918739366647233, 8.111399450257011173539533867234
