L(s) = 1 | + 3-s − 0.601i·5-s + 9-s + 0.163i·11-s + 3.37i·13-s − 0.601i·15-s − 2.13i·17-s + 0.953·19-s + 6.99i·23-s + 4.63·25-s + 27-s − 1.28·29-s + 3.35·31-s + 0.163i·33-s + 5.16·37-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.269i·5-s + 0.333·9-s + 0.0493i·11-s + 0.937i·13-s − 0.155i·15-s − 0.517i·17-s + 0.218·19-s + 1.45i·23-s + 0.927·25-s + 0.192·27-s − 0.239·29-s + 0.602·31-s + 0.0284i·33-s + 0.848·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.264129654\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.264129654\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 0.601iT - 5T^{2} \) |
| 11 | \( 1 - 0.163iT - 11T^{2} \) |
| 13 | \( 1 - 3.37iT - 13T^{2} \) |
| 17 | \( 1 + 2.13iT - 17T^{2} \) |
| 19 | \( 1 - 0.953T + 19T^{2} \) |
| 23 | \( 1 - 6.99iT - 23T^{2} \) |
| 29 | \( 1 + 1.28T + 29T^{2} \) |
| 31 | \( 1 - 3.35T + 31T^{2} \) |
| 37 | \( 1 - 5.16T + 37T^{2} \) |
| 41 | \( 1 - 6.67iT - 41T^{2} \) |
| 43 | \( 1 + 6.09iT - 43T^{2} \) |
| 47 | \( 1 + 3.69T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 - 8.05T + 59T^{2} \) |
| 61 | \( 1 - 0.181iT - 61T^{2} \) |
| 67 | \( 1 - 9.09iT - 67T^{2} \) |
| 71 | \( 1 - 3.36iT - 71T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 - 6.98iT - 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 14.8iT - 89T^{2} \) |
| 97 | \( 1 - 9.08iT - 97T^{2} \) |
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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
−8.989116838824046740593173398754, −8.402116995787331893767117370755, −7.42379777813853792056711700827, −6.94005473462300694097089038776, −5.88406194599786992179199162536, −4.96818067716932708022111172223, −4.17136606098289181855480017969, −3.24963041328052264143719738744, −2.25078439029949990646889721133, −1.12954921798893029611466155470,
0.855722908512736566870775507038, 2.31505538536968367248004785318, 3.04567783545437821169989824999, 3.99644063229530882532854614947, 4.90117792699665666875506268388, 5.87523544741277643540835792249, 6.69168700707592570941253504031, 7.46505941017333924986247508026, 8.312578932623884200112060387089, 8.722147523588057683775111441431