| L(s) = 1 | + i·3-s + (−0.353 + 2.20i)5-s − 3.09i·7-s − 9-s + 5.51·11-s − i·13-s + (−2.20 − 0.353i)15-s − 6.21i·17-s + 3.09·21-s − 4.21i·23-s + (−4.74 − 1.56i)25-s − i·27-s + 1.70·29-s + 5.51i·33-s + (6.83 + 1.09i)35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.158 + 0.987i)5-s − 1.16i·7-s − 0.333·9-s + 1.66·11-s − 0.277i·13-s + (−0.570 − 0.0913i)15-s − 1.50i·17-s + 0.675·21-s − 0.879i·23-s + (−0.949 − 0.312i)25-s − 0.192i·27-s + 0.317·29-s + 0.959i·33-s + (1.15 + 0.184i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.701894338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.701894338\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.353 - 2.20i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 + 3.09iT - 7T^{2} \) |
| 11 | \( 1 - 5.51T + 11T^{2} \) |
| 17 | \( 1 + 6.21iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 4.21iT - 23T^{2} \) |
| 29 | \( 1 - 1.70T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4.02iT - 37T^{2} \) |
| 41 | \( 1 - 0.198T + 41T^{2} \) |
| 43 | \( 1 - 5.70iT - 43T^{2} \) |
| 47 | \( 1 + 6.41iT - 47T^{2} \) |
| 53 | \( 1 + 4.02iT - 53T^{2} \) |
| 59 | \( 1 - 7.53T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 - 4.83iT - 67T^{2} \) |
| 71 | \( 1 - 7.98T + 71T^{2} \) |
| 73 | \( 1 + 9.31iT - 73T^{2} \) |
| 79 | \( 1 + 8.51T + 79T^{2} \) |
| 83 | \( 1 - 14.7iT - 83T^{2} \) |
| 89 | \( 1 - 9.40T + 89T^{2} \) |
| 97 | \( 1 - 14.3iT - 97T^{2} \) |
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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
−9.625854010483260752631334663528, −8.729258311305567762786716661888, −7.68251383408134281119923043187, −6.87335999721942848984820318009, −6.47791381763125457734911678478, −5.16445476879973813592626677767, −4.10802809226224230498329811639, −3.63020131342820948667985916823, −2.51223279918598086314823092119, −0.77535726508738285965449334217,
1.24421097952707926905615816444, 2.02844075721500231574512186600, 3.53822848536653462182597746547, 4.39303345259182666235874214740, 5.55465025012056404273194880529, 6.10773160957121925130823421425, 6.97271836101376287501848565498, 8.099482553472625764396436084101, 8.708296404523131894209151111794, 9.157454578737704674146628985372
