| L(s) = 1 | + (2 − 2i)3-s + (2 + 2i)7-s − 5i·9-s + (1 + i)13-s + (5 − 5i)17-s − 4·19-s + 8·21-s + (2 − 2i)23-s + (−4 − 4i)27-s + 4i·29-s − 4i·31-s + (−1 + i)37-s + 4·39-s + (−6 + 6i)43-s + (−2 − 2i)47-s + ⋯ |
| L(s) = 1 | + (1.15 − 1.15i)3-s + (0.755 + 0.755i)7-s − 1.66i·9-s + (0.277 + 0.277i)13-s + (1.21 − 1.21i)17-s − 0.917·19-s + 1.74·21-s + (0.417 − 0.417i)23-s + (−0.769 − 0.769i)27-s + 0.742i·29-s − 0.718i·31-s + (−0.164 + 0.164i)37-s + 0.640·39-s + (−0.914 + 0.914i)43-s + (−0.291 − 0.291i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.16068 - 1.20466i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.16068 - 1.20466i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2 + 2i)T - 3iT^{2} \) |
| 7 | \( 1 + (-2 - 2i)T + 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-1 - i)T + 13iT^{2} \) |
| 17 | \( 1 + (-5 + 5i)T - 17iT^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-2 + 2i)T - 23iT^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + (1 - i)T - 37iT^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + (6 - 6i)T - 43iT^{2} \) |
| 47 | \( 1 + (2 + 2i)T + 47iT^{2} \) |
| 53 | \( 1 + (-7 - 7i)T + 53iT^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + (10 + 10i)T + 67iT^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + (-3 - 3i)T + 73iT^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + (2 - 2i)T - 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (-3 + 3i)T - 97iT^{2} \) |
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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
−9.893377643754679860474305996525, −8.920870497237094801669080271307, −8.451078891250959854399863320179, −7.64253639911142376629320775619, −6.92319142458831143781788330724, −5.85994756202793568258382430565, −4.71431384262467874665852686988, −3.24478951700895614579558686438, −2.37495752714461338768708010697, −1.32326887397239178546294212757,
1.68588669912474800784623989980, 3.16453854697710220546237269357, 3.93607327561960163287629937021, 4.70361057410788386802092093070, 5.81953384772325011291471062080, 7.25470210770353773462164029968, 8.189337562407795786323827596724, 8.557331739005640020914052836296, 9.626181625712484079981184380971, 10.43811160386740641218102026062
