| L(s) = 1 | − 2.80i·5-s + 4.99i·7-s + 2.88·11-s + 6.56·13-s + 4.32i·17-s + 5.96i·19-s + 4.50·23-s − 2.89·25-s − 8.08i·29-s + 8.78i·31-s + 14.0·35-s − 37-s − 2.06i·41-s + 0.596i·43-s − 11.3·47-s + ⋯ |
| L(s) = 1 | − 1.25i·5-s + 1.88i·7-s + 0.868·11-s + 1.82·13-s + 1.04i·17-s + 1.36i·19-s + 0.939·23-s − 0.578·25-s − 1.50i·29-s + 1.57i·31-s + 2.37·35-s − 0.164·37-s − 0.323i·41-s + 0.0909i·43-s − 1.65·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.183555222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.183555222\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 + 2.80iT - 5T^{2} \) |
| 7 | \( 1 - 4.99iT - 7T^{2} \) |
| 11 | \( 1 - 2.88T + 11T^{2} \) |
| 13 | \( 1 - 6.56T + 13T^{2} \) |
| 17 | \( 1 - 4.32iT - 17T^{2} \) |
| 19 | \( 1 - 5.96iT - 19T^{2} \) |
| 23 | \( 1 - 4.50T + 23T^{2} \) |
| 29 | \( 1 + 8.08iT - 29T^{2} \) |
| 31 | \( 1 - 8.78iT - 31T^{2} \) |
| 41 | \( 1 + 2.06iT - 41T^{2} \) |
| 43 | \( 1 - 0.596iT - 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 9.48iT - 53T^{2} \) |
| 59 | \( 1 + 6.93T + 59T^{2} \) |
| 61 | \( 1 - 7.45T + 61T^{2} \) |
| 67 | \( 1 + 0.0130iT - 67T^{2} \) |
| 71 | \( 1 + 7.35T + 71T^{2} \) |
| 73 | \( 1 + 9.27T + 73T^{2} \) |
| 79 | \( 1 - 11.9iT - 79T^{2} \) |
| 83 | \( 1 - 1.66T + 83T^{2} \) |
| 89 | \( 1 + 15.4iT - 89T^{2} \) |
| 97 | \( 1 - 5.35T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.470937042068620179098125168541, −8.044625689721380338855099523923, −6.58437190307294402342943729125, −5.96906406450901130582981652662, −5.62828337837466297244914060363, −4.70862131622527079341125502296, −3.87205771383191038313262940907, −3.08051313707943178131364958123, −1.70840009762858937391365742831, −1.31879351215184879506473292589,
0.62984569808829105877017884475, 1.50742860332334738406686942996, 3.01057642483662825050101615140, 3.45730426849715335280551983467, 4.20170130827182545093136330184, 5.01035450987858706837502846555, 6.27384066927055111406766308304, 6.81630635909058262264534613995, 7.06022012322310798587385827963, 7.85724805409679198918757209014
