| L(s) = 1 | + i·2-s + i·3-s − 4-s − 3.45i·5-s − 6-s − i·8-s − 9-s + 3.45·10-s + (3.07 − 1.24i)11-s − i·12-s − 5.39·13-s + 3.45·15-s + 16-s + 0.0457·17-s − i·18-s − 6.95·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 1.54i·5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + 1.09·10-s + (0.927 − 0.373i)11-s − 0.288i·12-s − 1.49·13-s + 0.892·15-s + 0.250·16-s + 0.0110·17-s − 0.235i·18-s − 1.59·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3202763271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3202763271\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-3.07 + 1.24i)T \) |
| good | 5 | \( 1 + 3.45iT - 5T^{2} \) |
| 13 | \( 1 + 5.39T + 13T^{2} \) |
| 17 | \( 1 - 0.0457T + 17T^{2} \) |
| 19 | \( 1 + 6.95T + 19T^{2} \) |
| 23 | \( 1 - 1.19T + 23T^{2} \) |
| 29 | \( 1 + 7.78iT - 29T^{2} \) |
| 31 | \( 1 - 6.61iT - 31T^{2} \) |
| 37 | \( 1 + 1.51T + 37T^{2} \) |
| 41 | \( 1 - 4.87T + 41T^{2} \) |
| 43 | \( 1 - 3.27iT - 43T^{2} \) |
| 47 | \( 1 - 12.1iT - 47T^{2} \) |
| 53 | \( 1 + 3.69T + 53T^{2} \) |
| 59 | \( 1 - 14.8iT - 59T^{2} \) |
| 61 | \( 1 + 3.63T + 61T^{2} \) |
| 67 | \( 1 - 6.66T + 67T^{2} \) |
| 71 | \( 1 - 6.89T + 71T^{2} \) |
| 73 | \( 1 + 3.04T + 73T^{2} \) |
| 79 | \( 1 - 12.9iT - 79T^{2} \) |
| 83 | \( 1 + 8.48T + 83T^{2} \) |
| 89 | \( 1 - 4.20iT - 89T^{2} \) |
| 97 | \( 1 + 6.64iT - 97T^{2} \) |
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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
−9.016844187983622940186568608843, −8.364206248222383486089261947421, −7.71651225316120586671726508053, −6.67370032753938934039107464651, −5.92876545138558013835836565282, −5.12330931204386420204230165495, −4.45153578137853686420824703005, −4.06067620079175748679928773286, −2.58724882671701306148372530806, −1.18073755800012542634556486597,
0.10150204240798676230871261588, 1.87752184653140664702376192066, 2.40770857764418132578781120211, 3.31870906308012639696348068732, 4.15879423563547211410785118705, 5.15973109813240580939961491484, 6.27626359018639363151225551925, 6.86314418938148012730555886547, 7.35287185215621534557500144361, 8.273154864264554996683585651075
