| L(s) = 1 | − 2.56i·3-s + i·7-s − 3.56·9-s − 2.56·11-s − 4.56i·13-s − 4.56i·17-s + 1.12·19-s + 2.56·21-s − 5.12i·23-s + 1.43i·27-s + 5.68·29-s + 6.56i·33-s + 6i·37-s − 11.6·39-s − 3.12·41-s + ⋯ |
| L(s) = 1 | − 1.47i·3-s + 0.377i·7-s − 1.18·9-s − 0.772·11-s − 1.26i·13-s − 1.10i·17-s + 0.257·19-s + 0.558·21-s − 1.06i·23-s + 0.276i·27-s + 1.05·29-s + 1.14i·33-s + 0.986i·37-s − 1.87·39-s − 0.487·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9440200741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9440200741\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + 2.56iT - 3T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 + 4.56iT - 13T^{2} \) |
| 17 | \( 1 + 4.56iT - 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 + 5.12iT - 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 - 9.12iT - 43T^{2} \) |
| 47 | \( 1 + 3.68iT - 47T^{2} \) |
| 53 | \( 1 + 3.12iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 - 6.24iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 4.24iT - 73T^{2} \) |
| 79 | \( 1 + 6.56T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 7.12T + 89T^{2} \) |
| 97 | \( 1 + 14.8iT - 97T^{2} \) |
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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
−8.116557782474660725793978249388, −7.71307440425009204255983267927, −6.87258782661275900956477770744, −6.23078794631190053797573737869, −5.39636345351657888057026933719, −4.65738583788876981444789915797, −2.92953428440271108618337284769, −2.68528846344115209799596523772, −1.35076573566432879273001582546, −0.30399013352258424053596348510,
1.67465250342212139032783354021, 2.98029484433187708186807140340, 3.87603829961958412773186057119, 4.40014496932695222768942383595, 5.22072200494949401760437281947, 5.97889204769977280278178225125, 6.98341404500584579595330860166, 7.81923639343442653335817133977, 8.725952362379024111457395833808, 9.287313519232446368085767845799
