| L(s) = 1 | − i·2-s − 4-s + (1.94 − 1.10i)5-s − 1.41·7-s + i·8-s + (−1.10 − 1.94i)10-s + (0.772 + 3.22i)11-s + 3.62·13-s + 1.41i·14-s + 16-s − 2i·17-s − 6.07i·19-s + (−1.94 + 1.10i)20-s + (3.22 − 0.772i)22-s + 7.77·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.869 − 0.493i)5-s − 0.534·7-s + 0.353i·8-s + (−0.349 − 0.614i)10-s + (0.232 + 0.972i)11-s + 1.00·13-s + 0.377i·14-s + 0.250·16-s − 0.485i·17-s − 1.39i·19-s + (−0.434 + 0.246i)20-s + (0.687 − 0.164i)22-s + 1.62·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 990 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.31903 - 1.15090i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31903 - 1.15090i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.94 + 1.10i)T \) |
| 11 | \( 1 + (-0.772 - 3.22i)T \) |
| good | 7 | \( 1 + 1.41T + 7T^{2} \) |
| 13 | \( 1 - 3.62T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 6.07iT - 19T^{2} \) |
| 23 | \( 1 - 7.77T + 23T^{2} \) |
| 29 | \( 1 + 5.49T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 + 1.54iT - 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 - 9.45T + 43T^{2} \) |
| 47 | \( 1 - 9.96T + 47T^{2} \) |
| 53 | \( 1 + 6.07T + 53T^{2} \) |
| 59 | \( 1 + 0.794iT - 59T^{2} \) |
| 61 | \( 1 + 9.96iT - 61T^{2} \) |
| 67 | \( 1 + 3.08iT - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 - 4.06T + 73T^{2} \) |
| 79 | \( 1 + 6.07iT - 79T^{2} \) |
| 83 | \( 1 - 17.3iT - 83T^{2} \) |
| 89 | \( 1 - 11.4iT - 89T^{2} \) |
| 97 | \( 1 + 10.9iT - 97T^{2} \) |
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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
−9.584511619502020047297077493347, −9.302407978425320225844906674955, −8.503385318929513540349736062757, −7.14749022400745842020699260506, −6.38078226566663493600883108654, −5.23748373477274100219551977376, −4.54183660195868045140526981771, −3.26143969210946318964491603142, −2.21527591351315429488550917765, −0.964644529701851235319898058659,
1.33160661465551584868133741056, 3.03722319403263689252747693418, 3.84637806582439846172504130150, 5.33408472836088781989797956903, 6.08506785781258586422649252368, 6.51718864973088966454160735368, 7.58527428901224853308439428798, 8.627836463753594033261093434496, 9.151718244549913656509494212550, 10.17261511922698605991317462867
