| L(s) = 1 | + 3.24i·5-s + 1.84i·7-s + 5.77·11-s + 13-s − 3.60i·17-s + 0.235i·19-s − 8.93·23-s − 5.51·25-s + 4.24i·29-s − 2.62i·31-s − 5.99·35-s + 9.67·37-s + 6.07i·41-s + 6.08i·43-s + 12.1·47-s + ⋯ |
| L(s) = 1 | + 1.44i·5-s + 0.698i·7-s + 1.74·11-s + 0.277·13-s − 0.873i·17-s + 0.0540i·19-s − 1.86·23-s − 1.10·25-s + 0.787i·29-s − 0.471i·31-s − 1.01·35-s + 1.59·37-s + 0.948i·41-s + 0.927i·43-s + 1.76·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.047868139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.047868139\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.24iT - 5T^{2} \) |
| 7 | \( 1 - 1.84iT - 7T^{2} \) |
| 11 | \( 1 - 5.77T + 11T^{2} \) |
| 17 | \( 1 + 3.60iT - 17T^{2} \) |
| 19 | \( 1 - 0.235iT - 19T^{2} \) |
| 23 | \( 1 + 8.93T + 23T^{2} \) |
| 29 | \( 1 - 4.24iT - 29T^{2} \) |
| 31 | \( 1 + 2.62iT - 31T^{2} \) |
| 37 | \( 1 - 9.67T + 37T^{2} \) |
| 41 | \( 1 - 6.07iT - 41T^{2} \) |
| 43 | \( 1 - 6.08iT - 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 3.60iT - 53T^{2} \) |
| 59 | \( 1 + 3.16T + 59T^{2} \) |
| 61 | \( 1 + 3.41T + 61T^{2} \) |
| 67 | \( 1 - 4.23iT - 67T^{2} \) |
| 71 | \( 1 + 6.10T + 71T^{2} \) |
| 73 | \( 1 + 4.51T + 73T^{2} \) |
| 79 | \( 1 - 0.471iT - 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 8.89iT - 89T^{2} \) |
| 97 | \( 1 + 3.48T + 97T^{2} \) |
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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
−7.937414618878369453926668920820, −7.44370812291379199355096031616, −6.49091246188127636233511462484, −6.30989730308708523181481571434, −5.58434708854610226080352962128, −4.34076721330526136584294321452, −3.82503014576526259425141711906, −2.89914246617416415871075910280, −2.31594617483347835688505118250, −1.20291796871600476114713335022,
0.53640545580338433244383292902, 1.32980671321529673100508969869, 2.08101094211025505989064860098, 3.66561236059787272917941747317, 4.14199698552219726864062994844, 4.50934939557214641250750679661, 5.76673804581291135771778599240, 6.04536311070570877603769497668, 6.97220315158205338939365693397, 7.76493252494239049414958979973
