| L(s) = 1 | + (−2 − 2.23i)3-s + 2.23i·5-s + 6·7-s + (−1.00 + 8.94i)9-s − 4.47i·11-s − 16·13-s + (5.00 − 4.47i)15-s − 4.47i·17-s − 2·19-s + (−12 − 13.4i)21-s + 13.4i·23-s − 5.00·25-s + (22.0 − 15.6i)27-s − 31.3i·29-s + 18·31-s + ⋯ |
| L(s) = 1 | + (−0.666 − 0.745i)3-s + 0.447i·5-s + 0.857·7-s + (−0.111 + 0.993i)9-s − 0.406i·11-s − 1.23·13-s + (0.333 − 0.298i)15-s − 0.263i·17-s − 0.105·19-s + (−0.571 − 0.638i)21-s + 0.583i·23-s − 0.200·25-s + (0.814 − 0.579i)27-s − 1.07i·29-s + 0.580·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8057775848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8057775848\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2 + 2.23i)T \) |
| 5 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 - 6T + 49T^{2} \) |
| 11 | \( 1 + 4.47iT - 121T^{2} \) |
| 13 | \( 1 + 16T + 169T^{2} \) |
| 17 | \( 1 + 4.47iT - 289T^{2} \) |
| 19 | \( 1 + 2T + 361T^{2} \) |
| 23 | \( 1 - 13.4iT - 529T^{2} \) |
| 29 | \( 1 + 31.3iT - 841T^{2} \) |
| 31 | \( 1 - 18T + 961T^{2} \) |
| 37 | \( 1 - 16T + 1.36e3T^{2} \) |
| 41 | \( 1 + 62.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 16T + 1.84e3T^{2} \) |
| 47 | \( 1 + 49.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 4.47iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 4.47iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 82T + 3.72e3T^{2} \) |
| 67 | \( 1 - 24T + 4.48e3T^{2} \) |
| 71 | \( 1 + 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74T + 5.32e3T^{2} \) |
| 79 | \( 1 + 138T + 6.24e3T^{2} \) |
| 83 | \( 1 - 93.9iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 107. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 166T + 9.40e3T^{2} \) |
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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
−9.647372128490916177056163988783, −8.441782060796695905835553108148, −7.63957138570768393105015165377, −7.09512248957116142536995196827, −6.04742201197605889267125029810, −5.26097200966135114157771146720, −4.35289058879286594739913613427, −2.76107949072036508443570627110, −1.77046346568683217860950922347, −0.29129745499956941348431915741,
1.30583539631100481521339032542, 2.82205985169475438449777201734, 4.36979397075462679772016663981, 4.73333019269184443964324844352, 5.62391781855864904832379054048, 6.66990364275682517353426478945, 7.67910929315828346176679765769, 8.580149292440752593299018175974, 9.477230093787840282589283622605, 10.12801115097464981351042589267
