L(s) = 1 | − 1.58i·5-s − 7-s − 0.790i·11-s − 0.494i·13-s − 4.46·17-s + 7.55i·19-s + 0.839·23-s + 2.48·25-s − 2.76i·29-s + 0.568·31-s + 1.58i·35-s − 0.343i·37-s + 5.93·41-s + 3.16i·43-s + 4.80·47-s + ⋯ |
L(s) = 1 | − 0.708i·5-s − 0.377·7-s − 0.238i·11-s − 0.137i·13-s − 1.08·17-s + 1.73i·19-s + 0.175·23-s + 0.497·25-s − 0.513i·29-s + 0.102·31-s + 0.267i·35-s − 0.0564i·37-s + 0.926·41-s + 0.482i·43-s + 0.701·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.600663702\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.600663702\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.58iT - 5T^{2} \) |
| 11 | \( 1 + 0.790iT - 11T^{2} \) |
| 13 | \( 1 + 0.494iT - 13T^{2} \) |
| 17 | \( 1 + 4.46T + 17T^{2} \) |
| 19 | \( 1 - 7.55iT - 19T^{2} \) |
| 23 | \( 1 - 0.839T + 23T^{2} \) |
| 29 | \( 1 + 2.76iT - 29T^{2} \) |
| 31 | \( 1 - 0.568T + 31T^{2} \) |
| 37 | \( 1 + 0.343iT - 37T^{2} \) |
| 41 | \( 1 - 5.93T + 41T^{2} \) |
| 43 | \( 1 - 3.16iT - 43T^{2} \) |
| 47 | \( 1 - 4.80T + 47T^{2} \) |
| 53 | \( 1 - 4.36iT - 53T^{2} \) |
| 59 | \( 1 + 4.50iT - 59T^{2} \) |
| 61 | \( 1 + 5.40iT - 61T^{2} \) |
| 67 | \( 1 + 7.57iT - 67T^{2} \) |
| 71 | \( 1 - 9.52T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 3.71T + 79T^{2} \) |
| 83 | \( 1 - 7.30iT - 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 - 1.75T + 97T^{2} \) |
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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
−8.155952061646935005446364870503, −7.30403523123080901984632054070, −6.44317316881263847694428616192, −5.89093014975270527117936566679, −5.09750571463749261037443393110, −4.30463676412523097645012764755, −3.63333684981519697119432940709, −2.63152368694358757780294615407, −1.65059445942718988625445836743, −0.55545688929593120064673279021,
0.789636238363610403320788014861, 2.27636552126330435620473221295, 2.75506747842988269998559878056, 3.72636450679801537694263135281, 4.55662862589504539096554004496, 5.25117606280009473283019733804, 6.24504985044269282159730510262, 6.93237124856682055351384743551, 7.11914073658714352334878781847, 8.184900642484219340376704538996