| L(s) = 1 | + i·2-s − 4-s + (−1.44 + 1.70i)5-s + 4.38i·7-s − i·8-s + (−1.70 − 1.44i)10-s − 2·11-s + i·13-s − 4.38·14-s + 16-s + 5.86i·17-s + 0.973·19-s + (1.44 − 1.70i)20-s − 2i·22-s − 7.79i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.647 + 0.762i)5-s + 1.65i·7-s − 0.353i·8-s + (−0.538 − 0.457i)10-s − 0.603·11-s + 0.277i·13-s − 1.17·14-s + 0.250·16-s + 1.42i·17-s + 0.223·19-s + (0.323 − 0.381i)20-s − 0.426i·22-s − 1.62i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.647 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6720939554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6720939554\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.44 - 1.70i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 4.38iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 - 5.86iT - 17T^{2} \) |
| 19 | \( 1 - 0.973T + 19T^{2} \) |
| 23 | \( 1 + 7.79iT - 23T^{2} \) |
| 29 | \( 1 - 0.973T + 29T^{2} \) |
| 31 | \( 1 + 1.79T + 31T^{2} \) |
| 37 | \( 1 - 0.591iT - 37T^{2} \) |
| 41 | \( 1 + 4.81T + 41T^{2} \) |
| 43 | \( 1 - 4.68iT - 43T^{2} \) |
| 47 | \( 1 - 0.381iT - 47T^{2} \) |
| 53 | \( 1 + 7.79iT - 53T^{2} \) |
| 59 | \( 1 + 0.973T + 59T^{2} \) |
| 61 | \( 1 + 0.817T + 61T^{2} \) |
| 67 | \( 1 + 1.79iT - 67T^{2} \) |
| 71 | \( 1 - 3.92T + 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 6.97iT - 83T^{2} \) |
| 89 | \( 1 - 0.973T + 89T^{2} \) |
| 97 | \( 1 - 18.6iT - 97T^{2} \) |
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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
−10.28521929157350659700585512083, −9.284412370839344464965165106663, −8.305493564549353380403767203993, −8.133595025383295160024115066909, −6.82351808979859189029211618657, −6.23325433754359786181471949250, −5.39120235094944819730129591926, −4.37557459772107776443062557125, −3.19936425922261964593591028157, −2.19872716927406850730121821739,
0.30615968849467756538030730029, 1.36293244205502631529179754541, 3.08300638830055572680234468864, 3.90216728695400336197125854639, 4.74749992766504637091983227286, 5.49919235221209091286123473239, 7.23090254838674317138751702169, 7.49990741052037150857592505252, 8.476049687382180420013870595392, 9.468624312359301737651838718975
