L(s) = 1 | + (0.707 + 0.707i)3-s + (−1.87 − 1.87i)7-s + 1.00i·9-s − 2·11-s + (1.41 + 1.41i)13-s + (2.82 − 2.82i)17-s − 5.29·19-s − 2.64i·21-s + (−3.74 + 3.74i)23-s + (−0.707 + 0.707i)27-s − 6i·29-s − 5.29i·31-s + (−1.41 − 1.41i)33-s + 2.00i·39-s − 10.5i·41-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.707 − 0.707i)7-s + 0.333i·9-s − 0.603·11-s + (0.392 + 0.392i)13-s + (0.685 − 0.685i)17-s − 1.21·19-s − 0.577i·21-s + (−0.780 + 0.780i)23-s + (−0.136 + 0.136i)27-s − 1.11i·29-s − 0.950i·31-s + (−0.246 − 0.246i)33-s + 0.320i·39-s − 1.65i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7257566093\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7257566093\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.87 + 1.87i)T \) |
good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + (-1.41 - 1.41i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.82 + 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 + (3.74 - 3.74i)T - 23iT^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 5.29iT - 31T^{2} \) |
| 37 | \( 1 + 37iT^{2} \) |
| 41 | \( 1 + 10.5iT - 41T^{2} \) |
| 43 | \( 1 + (-3.74 + 3.74i)T - 43iT^{2} \) |
| 47 | \( 1 + (5.65 - 5.65i)T - 47iT^{2} \) |
| 53 | \( 1 + (3.74 - 3.74i)T - 53iT^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 10.5iT - 61T^{2} \) |
| 67 | \( 1 + (3.74 + 3.74i)T + 67iT^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + (9.89 + 9.89i)T + 73iT^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + (1.41 - 1.41i)T - 97iT^{2} \) |
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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
−9.032115882315453323223827469639, −7.893436710241932003650516746433, −7.54431803293349403765259862829, −6.41795932206106994328723491731, −5.75656938778817031452511481804, −4.57537339521422400042105331441, −3.89933737968782220526082332829, −3.04507262960608600060118068245, −1.94880145375329355425806551767, −0.22883945314484038727294940400,
1.52194427612685926599770805903, 2.67279158466802780000981332975, 3.34010248526607017740878040589, 4.46629800566216696214798269834, 5.59967274963354355920854064333, 6.24538047368015423792902146907, 6.94766850362589726666134204254, 8.105580610999581757681200165043, 8.410514155606214802309069114949, 9.232855993903868969781222800982