L(s) = 1 | − 5-s + 0.754i·7-s + 2.93·11-s − 1.92·13-s − 5.65·17-s + 0.739i·19-s + (−1.74 + 4.46i)23-s + 25-s + 8.80i·29-s − 10.6·31-s − 0.754i·35-s − 4.79i·37-s − 6.47i·41-s + 0.433i·43-s − 7.46i·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.285i·7-s + 0.885·11-s − 0.534·13-s − 1.37·17-s + 0.169i·19-s + (−0.364 + 0.931i)23-s + 0.200·25-s + 1.63i·29-s − 1.90·31-s − 0.127i·35-s − 0.788i·37-s − 1.01i·41-s + 0.0660i·43-s − 1.08i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9691750441\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9691750441\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + (1.74 - 4.46i)T \) |
good | 7 | \( 1 - 0.754iT - 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 + 1.92T + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 - 0.739iT - 19T^{2} \) |
| 29 | \( 1 - 8.80iT - 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 + 4.79iT - 37T^{2} \) |
| 41 | \( 1 + 6.47iT - 41T^{2} \) |
| 43 | \( 1 - 0.433iT - 43T^{2} \) |
| 47 | \( 1 + 7.46iT - 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 8.48iT - 59T^{2} \) |
| 61 | \( 1 + 10.0iT - 61T^{2} \) |
| 67 | \( 1 - 7.76iT - 67T^{2} \) |
| 71 | \( 1 + 3.47iT - 71T^{2} \) |
| 73 | \( 1 + 1.66T + 73T^{2} \) |
| 79 | \( 1 + 1.66iT - 79T^{2} \) |
| 83 | \( 1 - 2.45T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 - 15.9iT - 97T^{2} \) |
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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
−7.35440545899137516930399508693, −7.19374427765756729053324155399, −6.36572459301706164321525891823, −5.47835649030776790307647835928, −4.95556966345461469143911912940, −3.83440809191185039478959115955, −3.65201721793778621922429209766, −2.33965478561276367531328199240, −1.68230410422716836366035673467, −0.27907873771996920627414792720,
0.822908025965177289578869466609, 2.04201041873978737558565590755, 2.76251839774803403829225242011, 3.97255441577119237402834766017, 4.21443187646531820109735015624, 5.03667321248485489099440646078, 6.07071040143288278289118030995, 6.56144869593460241248856923836, 7.32471090924503851221919823786, 7.80291170121813472899493405280