| L(s) = 1 | − i·2-s + i·3-s − 4-s − 4.05i·5-s + 6-s + i·8-s − 9-s − 4.05·10-s + (−1.17 − 3.09i)11-s − i·12-s + 3.91·13-s + 4.05·15-s + 16-s − 0.188·17-s + i·18-s − 4.37·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 1.81i·5-s + 0.408·6-s + 0.353i·8-s − 0.333·9-s − 1.28·10-s + (−0.355 − 0.934i)11-s − 0.288i·12-s + 1.08·13-s + 1.04·15-s + 0.250·16-s − 0.0456·17-s + 0.235i·18-s − 1.00·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7199654687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7199654687\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.17 + 3.09i)T \) |
| good | 5 | \( 1 + 4.05iT - 5T^{2} \) |
| 13 | \( 1 - 3.91T + 13T^{2} \) |
| 17 | \( 1 + 0.188T + 17T^{2} \) |
| 19 | \( 1 + 4.37T + 19T^{2} \) |
| 23 | \( 1 + 3.58T + 23T^{2} \) |
| 29 | \( 1 + 5.75iT - 29T^{2} \) |
| 31 | \( 1 - 3.21iT - 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 + 8.14T + 41T^{2} \) |
| 43 | \( 1 + 3.21iT - 43T^{2} \) |
| 47 | \( 1 + 4.37iT - 47T^{2} \) |
| 53 | \( 1 - 2.83T + 53T^{2} \) |
| 59 | \( 1 + 4.99iT - 59T^{2} \) |
| 61 | \( 1 - 7.95T + 61T^{2} \) |
| 67 | \( 1 + 6.50T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 8.55iT - 79T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 - 14.2iT - 89T^{2} \) |
| 97 | \( 1 - 17.4iT - 97T^{2} \) |
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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
−8.408352577043288710247879286967, −8.052439831322302383881839277767, −6.38504074560811410882392100750, −5.61240798603598464815944361290, −5.03543749357220788676054508661, −4.10207243319156096879082350664, −3.71573259922501208033026771910, −2.34893769834423160681441820120, −1.23743758133423985525554702779, −0.22826896881159961243742488545,
1.75110771828002616911158885457, 2.69206444165601744704562657619, 3.60358660427476249036309664089, 4.47757816294904126015100365782, 5.76616760139211036301060493264, 6.28149054014169147199450096093, 6.89417023744843167269212847292, 7.43911677524007157756940871322, 8.120651071713730734261998609559, 8.896240613991135137306140656370
