| L(s) = 1 | + i·2-s + 1.84i·3-s − 4-s − 2.47i·5-s − 1.84·6-s − i·8-s − 0.414·9-s + 2.47·10-s + (−1.99 + 2.65i)11-s − 1.84i·12-s + 1.96·13-s + 4.56·15-s + 16-s + 5.64·17-s − 0.414i·18-s + 0.906·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 1.06i·3-s − 0.5·4-s − 1.10i·5-s − 0.754·6-s − 0.353i·8-s − 0.138·9-s + 0.781·10-s + (−0.600 + 0.799i)11-s − 0.533i·12-s + 0.544·13-s + 1.17·15-s + 0.250·16-s + 1.37·17-s − 0.0976i·18-s + 0.208·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.515651505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.515651505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (1.99 - 2.65i)T \) |
| good | 3 | \( 1 - 1.84iT - 3T^{2} \) |
| 5 | \( 1 + 2.47iT - 5T^{2} \) |
| 13 | \( 1 - 1.96T + 13T^{2} \) |
| 17 | \( 1 - 5.64T + 17T^{2} \) |
| 19 | \( 1 - 0.906T + 19T^{2} \) |
| 23 | \( 1 + 0.936T + 23T^{2} \) |
| 29 | \( 1 - 7.50iT - 29T^{2} \) |
| 31 | \( 1 - 4.71iT - 31T^{2} \) |
| 37 | \( 1 + 9.62T + 37T^{2} \) |
| 41 | \( 1 - 6.93T + 41T^{2} \) |
| 43 | \( 1 - 6.80iT - 43T^{2} \) |
| 47 | \( 1 + 4.00iT - 47T^{2} \) |
| 53 | \( 1 - 6.56T + 53T^{2} \) |
| 59 | \( 1 + 0.965iT - 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 6.72T + 67T^{2} \) |
| 71 | \( 1 + 11.5T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 8.48iT - 79T^{2} \) |
| 83 | \( 1 - 4.83T + 83T^{2} \) |
| 89 | \( 1 + 12.7iT - 89T^{2} \) |
| 97 | \( 1 - 3.82iT - 97T^{2} \) |
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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
−10.11501729094403439568614562930, −9.185154142584223473900806768928, −8.670590260937087125280685165479, −7.74137736745695877765363505959, −6.88393667997065836345989555586, −5.43682004220998240126627298786, −5.15307327928306553638314584435, −4.25957693173666839395289730265, −3.32847530633674068522511184755, −1.31380076510262788926795596808,
0.78862876064001558754242661617, 2.13819699158310587612649567526, 3.03833891844947174397787597508, 3.94313263481047059996231023657, 5.54568159263050625194260018011, 6.20571491013444669688979428556, 7.26194970868531987879879761254, 7.82895892559961164132300377147, 8.681395246520751704553844388949, 9.949817922512495015787575742987
