| L(s) = 1 | + 3.16i·3-s + 4.24·7-s − 7.00·9-s + 13.4i·21-s − 1.41·23-s − 12.6i·27-s + 8.94i·29-s − 12·41-s + 3.16i·43-s − 9.89·47-s + 10.9·49-s + 13.4i·61-s − 29.6·63-s + 15.8i·67-s − 4.47i·69-s + ⋯ |
| L(s) = 1 | + 1.82i·3-s + 1.60·7-s − 2.33·9-s + 2.92i·21-s − 0.294·23-s − 2.43i·27-s + 1.66i·29-s − 1.87·41-s + 0.482i·43-s − 1.44·47-s + 1.57·49-s + 1.71i·61-s − 3.74·63-s + 1.93i·67-s − 0.538i·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.614668560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.614668560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3.16iT - 3T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 - 8.94iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 - 3.16iT - 43T^{2} \) |
| 47 | \( 1 + 9.89T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 13.4iT - 61T^{2} \) |
| 67 | \( 1 - 15.8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 9.48iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.927024866643178444056521243226, −8.547031697008030689501281972201, −7.84616596504854576412816619281, −6.75220449730484085054574464100, −5.52956896075894788336641543047, −5.11023260360190335735175600501, −4.48218401815191520097636881223, −3.73833360125606421002503620570, −2.82051180885537447940440090741, −1.53881792470247520344419201980,
0.47868859532977740265935942113, 1.71702779871546634814321851182, 2.03394137902881336513794194195, 3.27172293108788720407437899697, 4.60297674271971492224693455695, 5.36549483512387828763251084359, 6.20362314204847542444611779041, 6.86313756113032677674797398866, 7.71017836851461426821301897121, 8.101917750529173277558681843698
