| L(s) = 1 | + (−0.707 + 2.12i)5-s + 6i·13-s + 7.07·17-s + (−3.99 − 3i)25-s + 4.24i·29-s − 12i·37-s + 12.7i·41-s − 7·49-s + 7.07·53-s − 10·61-s + (−12.7 − 4.24i)65-s + 6i·73-s + (−5.00 + 15i)85-s + 4.24i·89-s + 18i·97-s + ⋯ |
| L(s) = 1 | + (−0.316 + 0.948i)5-s + 1.66i·13-s + 1.71·17-s + (−0.799 − 0.600i)25-s + 0.787i·29-s − 1.97i·37-s + 1.98i·41-s − 49-s + 0.971·53-s − 1.28·61-s + (−1.57 − 0.526i)65-s + 0.702i·73-s + (−0.542 + 1.62i)85-s + 0.449i·89-s + 1.82i·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.371155713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.371155713\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 2.12i)T \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 7.07T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 12iT - 37T^{2} \) |
| 41 | \( 1 - 12.7iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 4.24iT - 89T^{2} \) |
| 97 | \( 1 - 18iT - 97T^{2} \) |
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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
−9.187273287356262902412704130340, −8.116816809326193751713237691755, −7.49863767741096796277037602839, −6.79914013141463909711840572351, −6.14309330691393924884483483130, −5.20283243349634138889503424427, −4.16150317739541778027866496938, −3.47608607859017620415163371396, −2.53389640394113258392997818017, −1.40252287142828062704649878897,
0.46669339364805702201625502317, 1.45743184667261608486886134098, 2.93501693731236787527533473953, 3.64961831676758399020587630265, 4.70190782242523148477304023285, 5.45367332801539869310839033897, 5.94121294461546479036682564835, 7.21093873457968363358557343569, 7.980215464580651931237380549655, 8.268606143437232781847687158371
