| L(s) = 1 | − i·2-s − 4-s − 5-s + (−1.10 + 2.40i)7-s + i·8-s + i·10-s − i·11-s − 3.01i·13-s + (2.40 + 1.10i)14-s + 16-s − 6.21·17-s + 3.22i·19-s + 20-s − 22-s + 4.17i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.447·5-s + (−0.418 + 0.908i)7-s + 0.353i·8-s + 0.316i·10-s − 0.301i·11-s − 0.836i·13-s + (0.642 + 0.296i)14-s + 0.250·16-s − 1.50·17-s + 0.739i·19-s + 0.223·20-s − 0.213·22-s + 0.870i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.006870578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.006870578\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + (1.10 - 2.40i)T \) |
| 11 | \( 1 + iT \) |
| good | 13 | \( 1 + 3.01iT - 13T^{2} \) |
| 17 | \( 1 + 6.21T + 17T^{2} \) |
| 19 | \( 1 - 3.22iT - 19T^{2} \) |
| 23 | \( 1 - 4.17iT - 23T^{2} \) |
| 29 | \( 1 - 1.82iT - 29T^{2} \) |
| 31 | \( 1 + 9.63iT - 31T^{2} \) |
| 37 | \( 1 + 2.87T + 37T^{2} \) |
| 41 | \( 1 + 3.16T + 41T^{2} \) |
| 43 | \( 1 - 0.359T + 43T^{2} \) |
| 47 | \( 1 + 0.578T + 47T^{2} \) |
| 53 | \( 1 - 5.30iT - 53T^{2} \) |
| 59 | \( 1 - 4.27T + 59T^{2} \) |
| 61 | \( 1 + 6.80iT - 61T^{2} \) |
| 67 | \( 1 - 2.10T + 67T^{2} \) |
| 71 | \( 1 - 6.45iT - 71T^{2} \) |
| 73 | \( 1 - 3.37iT - 73T^{2} \) |
| 79 | \( 1 + 2.13T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 + 9.79T + 89T^{2} \) |
| 97 | \( 1 + 7.51iT - 97T^{2} \) |
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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
−8.043236169977056862217767583138, −7.23229028208964201499283756692, −6.28074308337281609104594833677, −5.68712272289974906965312374615, −4.95846235966961275504589511831, −4.02925119423402244720414517892, −3.38283476741470944770688905962, −2.59223638973532965149735017424, −1.81170880389044763559571063070, −0.43751258280740362654585537563,
0.56679958714579571046835964292, 1.90565929729474621585054818048, 3.04915148813015767878196068939, 3.97259245407488042942311558749, 4.53036986092092389847330015569, 5.06838537693220686505183939824, 6.32012072897708319680770621542, 6.84998377922411755828432722587, 7.06164577697639859229593674783, 8.004712101570352842237850769189
