| L(s) = 1 | + i·3-s + (1.41 − 1.73i)5-s − 9-s + 3.46i·13-s + (1.73 + 1.41i)15-s + 4.89i·17-s − 4.89·19-s + 2.82i·23-s + (−0.999 − 4.89i)25-s − i·27-s + 2.82·29-s − 6.92·31-s + 3.46i·37-s − 3.46·39-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.632 − 0.774i)5-s − 0.333·9-s + 0.960i·13-s + (0.447 + 0.365i)15-s + 1.18i·17-s − 1.12·19-s + 0.589i·23-s + (−0.199 − 0.979i)25-s − 0.192i·27-s + 0.525·29-s − 1.24·31-s + 0.569i·37-s − 0.554·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.007567089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.007567089\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1.41 + 1.73i)T \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 4.89T + 19T^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 3.46iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 2.82iT - 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 9.79T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 9.79iT - 97T^{2} \) |
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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
−8.925338145470165814720169264652, −8.303741009673758852895747592272, −7.32644305062341832582260367214, −6.30002928849844835567139395393, −5.87185321713164987140421725809, −4.89409936084408825171795624197, −4.31045729127959837066588738023, −3.52381709064467841593341870691, −2.21289561185688581022160480524, −1.46159467372924295444520856780,
0.27023378890776673211585483243, 1.69374138597933502718447798230, 2.60910678529499473205078867779, 3.21614625735924866595640050955, 4.42247595979907507043026913044, 5.45431216119313614317958784943, 5.96829990721161535182934569509, 6.89557290186459924412970427214, 7.23041572485089103241778400060, 8.179936055341155661729530877146
