| L(s) = 1 | − 2i·3-s + (1 − 2i)5-s − 2i·7-s − 9-s − 4·11-s + 4i·13-s + (−4 − 2i)15-s + 4·19-s − 4·21-s − 2i·23-s + (−3 − 4i)25-s − 4i·27-s + 2·29-s + 8i·33-s + (−4 − 2i)35-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + (0.447 − 0.894i)5-s − 0.755i·7-s − 0.333·9-s − 1.20·11-s + 1.10i·13-s + (−1.03 − 0.516i)15-s + 0.917·19-s − 0.872·21-s − 0.417i·23-s + (−0.600 − 0.800i)25-s − 0.769i·27-s + 0.371·29-s + 1.39i·33-s + (−0.676 − 0.338i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.689985 - 1.11642i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.689985 - 1.11642i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−11.56003040110246279599858175777, −10.35601431790166315181169347920, −9.442278828225751355671888691640, −8.286652205325374798821505754513, −7.49390492065617882517364478080, −6.61140314742272237450934732165, −5.41765569838809309218604951719, −4.27704939201964139077046919386, −2.33249497133181333909952594374, −1.00408633879289146220868726536,
2.57721743382362538607804729472, 3.51247017092291311420608862381, 5.15679754884998928140906335902, 5.67291059682368933672061423358, 7.16092932639731528465211978098, 8.244666969700496587120338990414, 9.439291448262045443523774789712, 10.18685347995366398591103840290, 10.67417227178092479248836645418, 11.70788594371610619857230734194
