| L(s) = 1 | + (−1.41 − 1.41i)5-s − 7-s + (−1.41 + 1.41i)11-s + (2 + 2i)13-s + 7.07i·23-s − 0.999i·25-s + (1.41 − 1.41i)29-s − 8i·31-s + (1.41 + 1.41i)35-s + (5 − 5i)37-s − 2.82·41-s + (3 + 3i)43-s − 5.65·47-s + 49-s + 4.00·55-s + ⋯ |
| L(s) = 1 | + (−0.632 − 0.632i)5-s − 0.377·7-s + (−0.426 + 0.426i)11-s + (0.554 + 0.554i)13-s + 1.47i·23-s − 0.199i·25-s + (0.262 − 0.262i)29-s − 1.43i·31-s + (0.239 + 0.239i)35-s + (0.821 − 0.821i)37-s − 0.441·41-s + (0.457 + 0.457i)43-s − 0.825·47-s + 0.142·49-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.533 + 0.845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7764587500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7764587500\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.41 + 1.41i)T + 5iT^{2} \) |
| 11 | \( 1 + (1.41 - 1.41i)T - 11iT^{2} \) |
| 13 | \( 1 + (-2 - 2i)T + 13iT^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19iT^{2} \) |
| 23 | \( 1 - 7.07iT - 23T^{2} \) |
| 29 | \( 1 + (-1.41 + 1.41i)T - 29iT^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 + (-5 + 5i)T - 37iT^{2} \) |
| 41 | \( 1 + 2.82T + 41T^{2} \) |
| 43 | \( 1 + (-3 - 3i)T + 43iT^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + (-2.82 + 2.82i)T - 59iT^{2} \) |
| 61 | \( 1 + (2 + 2i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5 + 5i)T - 67iT^{2} \) |
| 71 | \( 1 + 9.89iT - 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 16iT - 79T^{2} \) |
| 83 | \( 1 + (-2.82 - 2.82i)T + 83iT^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−8.016604862119424109799126949955, −7.69681560425415482193891161424, −6.73378638269895212653257551211, −5.97104767417856032590609226922, −5.16441451975991953119885012224, −4.27748904772342348833274011733, −3.75194804215127376903277288291, −2.63940476403183025939586073807, −1.53546023301149047920337229545, −0.25493654303091938193168758860,
1.07299328130392228185385731261, 2.61876539329111922241849947472, 3.19420525361353250450356385950, 3.98700823947144538083780697476, 4.95901799655736879985017601878, 5.78683107609330705098009086812, 6.65307498322822593018259630051, 7.07923566349742673772959419382, 8.182242013684582520005207799946, 8.387710738716455362458750194874
