| L(s) = 1 | + 89.8·5-s + 443.·7-s + 42.6·11-s − 1.28e3i·13-s + 6.98e3i·17-s + 5.87e3i·19-s + 4.67e3i·23-s − 7.54e3·25-s − 2.90e4·29-s − 3.59e4·31-s + 3.98e4·35-s + 1.34e4i·37-s + 9.43e4i·41-s + 1.27e5i·43-s + 2.35e4i·47-s + ⋯ |
| L(s) = 1 | + 0.718·5-s + 1.29·7-s + 0.0320·11-s − 0.583i·13-s + 1.42i·17-s + 0.856i·19-s + 0.384i·23-s − 0.483·25-s − 1.18·29-s − 1.20·31-s + 0.929·35-s + 0.266i·37-s + 1.36i·41-s + 1.59i·43-s + 0.227i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.346 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.028022724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.028022724\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 89.8T + 1.56e4T^{2} \) |
| 7 | \( 1 - 443.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 42.6T + 1.77e6T^{2} \) |
| 13 | \( 1 + 1.28e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 6.98e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.87e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 4.67e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 2.90e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 3.59e4T + 8.87e8T^{2} \) |
| 37 | \( 1 - 1.34e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 9.43e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.27e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 - 2.35e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 5.07e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 2.51e5T + 4.21e10T^{2} \) |
| 61 | \( 1 - 4.41e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 2.75e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.02e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.26e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 1.32e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 3.36e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 4.87e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 5.48e5T + 8.32e11T^{2} \) |
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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
−10.07772858077445278344037948920, −9.189806635236010945432155066436, −8.099959579820094473160621029336, −7.66890523441825232003483724967, −6.16499613593295505824042173940, −5.58464781181447661549669719675, −4.53379561638642171777605038876, −3.42158699095809895670420569688, −1.90429950957063435902506380460, −1.41403640188329189079288336418,
0.36014164954849619559866115500, 1.69769043433377057651016616460, 2.39572882611158899224598514293, 3.93140700634153748737295822653, 5.01792608322146132134559212169, 5.61367607035301386284155513695, 6.96185911800633359095357301740, 7.58783491794073712807205140469, 8.853392399071606536584808007073, 9.302708910261240796950829929349
