L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (2.12 + 0.707i)5-s + (−0.707 − 0.707i)8-s + (2 − 0.999i)10-s + 2.82i·11-s + (−3 + 3i)13-s − 1.00·16-s + (2.82 − 2.82i)17-s − 4i·19-s + (0.707 − 2.12i)20-s + (2.00 + 2.00i)22-s + (−0.707 − 0.707i)23-s + (3.99 + 3i)25-s + 4.24i·26-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.948 + 0.316i)5-s + (−0.250 − 0.250i)8-s + (0.632 − 0.316i)10-s + 0.852i·11-s + (−0.832 + 0.832i)13-s − 0.250·16-s + (0.685 − 0.685i)17-s − 0.917i·19-s + (0.158 − 0.474i)20-s + (0.426 + 0.426i)22-s + (−0.147 − 0.147i)23-s + (0.799 + 0.600i)25-s + 0.832i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.804246128\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.804246128\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.12 - 0.707i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
good | 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.82 + 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 29 | \( 1 - 9.89T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (-7 - 7i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (-2 + 2i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.82 - 2.82i)T - 47iT^{2} \) |
| 53 | \( 1 + (-8.48 - 8.48i)T + 53iT^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + (-10 - 10i)T + 67iT^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (-1 + i)T - 73iT^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + (8.48 + 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.89T + 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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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
−9.395920775107525535068607589526, −8.453247462135531114001097686862, −7.16791969622481918160130029003, −6.75168062269913017640962593562, −5.81619907731749240020066906539, −4.83964477779042021106035940591, −4.41233228292337686585482992965, −2.83072815466402933564986921391, −2.44345064198214491609156656820, −1.16055218300553806200644341881,
1.02986406041960055157200638413, 2.49340188267038863365719764576, 3.31014646632361881496256017078, 4.51448506179189041368230527320, 5.26542487366916744802791702977, 6.08029868518721278892245931567, 6.41765892315836364799264553632, 7.85801542369700998923062862754, 8.099400030953032235504033687247, 9.099118695181456932068518202522