| L(s) = 1 | + 0.910·3-s + 6.34·5-s − 8.17·9-s − 13.2i·11-s + 19.4·13-s + 5.77·15-s − 15.9i·17-s + 16.4·19-s − 23.9·23-s + 15.2·25-s − 15.6·27-s + 16.6i·29-s − 12.8i·31-s − 12.0i·33-s − 47.5i·37-s + ⋯ |
| L(s) = 1 | + 0.303·3-s + 1.26·5-s − 0.907·9-s − 1.20i·11-s + 1.49·13-s + 0.385·15-s − 0.936i·17-s + 0.866·19-s − 1.04·23-s + 0.610·25-s − 0.579·27-s + 0.574i·29-s − 0.414i·31-s − 0.364i·33-s − 1.28i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.427 + 0.904i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.427 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.665287677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.665287677\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.910T + 9T^{2} \) |
| 5 | \( 1 - 6.34T + 25T^{2} \) |
| 11 | \( 1 + 13.2iT - 121T^{2} \) |
| 13 | \( 1 - 19.4T + 169T^{2} \) |
| 17 | \( 1 + 15.9iT - 289T^{2} \) |
| 19 | \( 1 - 16.4T + 361T^{2} \) |
| 23 | \( 1 + 23.9T + 529T^{2} \) |
| 29 | \( 1 - 16.6iT - 841T^{2} \) |
| 31 | \( 1 + 12.8iT - 961T^{2} \) |
| 37 | \( 1 + 47.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 6.49iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 33.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 21.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 37.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 54.6T + 3.48e3T^{2} \) |
| 61 | \( 1 + 10.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 17.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 32.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 107. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 58.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 36.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 1.07iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 169. iT - 9.40e3T^{2} \) |
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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
−9.035258018561191765774353132560, −8.536451402022278569511655178467, −7.62359233728541331876219723337, −6.39116911276232000476613069439, −5.81344604912648816297609036761, −5.32157159528192842762297871258, −3.75558698790115334150654093116, −2.99392619643099752402690910930, −1.95920400044620717928056075031, −0.69379654989450347744180622318,
1.38025841902953580758394298664, 2.18987767049542325331292743052, 3.27897722749198322512990106937, 4.32552212759889575544143185655, 5.54669701097010997833990753801, 6.00738307482141257233516822825, 6.81829142053750450346467442997, 8.030338987171759130851928454794, 8.563160085564018492133283475751, 9.518892811745109079058879433358
