| L(s) = 1 | + 47.0i·3-s − 22.4·5-s + 196.·7-s − 1.48e3·9-s − 819.·11-s + 3.59e3i·13-s − 1.05e3i·15-s + 605.·17-s + (−3.01e3 − 6.16e3i)19-s + 9.22e3i·21-s + 3.75e3·23-s − 1.51e4·25-s − 3.56e4i·27-s − 1.14e4i·29-s − 8.37e3i·31-s + ⋯ |
| L(s) = 1 | + 1.74i·3-s − 0.179·5-s + 0.571·7-s − 2.03·9-s − 0.615·11-s + 1.63i·13-s − 0.313i·15-s + 0.123·17-s + (−0.439 − 0.898i)19-s + 0.996i·21-s + 0.309·23-s − 0.967·25-s − 1.81i·27-s − 0.470i·29-s − 0.281i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.04764581628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04764581628\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (3.01e3 + 6.16e3i)T \) |
| good | 3 | \( 1 - 47.0iT - 729T^{2} \) |
| 5 | \( 1 + 22.4T + 1.56e4T^{2} \) |
| 7 | \( 1 - 196.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 819.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 3.59e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 605.T + 2.41e7T^{2} \) |
| 23 | \( 1 - 3.75e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + 1.14e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 8.37e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 3.57e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 7.24e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 9.34e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 3.89e4T + 1.07e10T^{2} \) |
| 53 | \( 1 - 7.03e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.37e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 6.37e4T + 5.15e10T^{2} \) |
| 67 | \( 1 + 5.63e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 5.72e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 4.40e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 1.01e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 6.96e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 7.67e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 5.28e5iT - 8.32e11T^{2} \) |
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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
−10.52863653188100269442699854470, −9.562598485450341364088428202573, −8.922050449019566260605602639731, −7.893567315928157780516686418913, −6.43908756375912594809862008199, −5.08747706944055763031587965948, −4.50474080951313445441650445807, −3.53663535736961866010855554869, −2.13688439822967660156580005868, −0.01198153347700335812507247197,
1.08973963486166837462271617237, 2.14035817823538984676330537035, 3.30010436776855859024900576987, 5.19495674171324922838109340988, 6.02758128975759489015649420344, 7.16197558283151276923255510126, 8.063545935945660362622141250351, 8.294483672967009185791193289803, 10.04593676465160245864660909594, 11.08703999098736722813424812591
