L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + (3.23 − 3.81i)5-s + (−8.15 − 4.70i)7-s + 2.82·8-s + (2.37 + 6.65i)10-s + (−2.03 − 1.17i)11-s + (−1.33 + 0.769i)13-s + (11.5 − 6.65i)14-s + (−2.00 + 3.46i)16-s − 11.0·17-s + 7.09·19-s + (−9.83 − 1.79i)20-s + (2.88 − 1.66i)22-s + (−4.09 − 7.09i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.647 − 0.762i)5-s + (−1.16 − 0.672i)7-s + 0.353·8-s + (0.237 + 0.665i)10-s + (−0.185 − 0.107i)11-s + (−0.102 + 0.0591i)13-s + (0.823 − 0.475i)14-s + (−0.125 + 0.216i)16-s − 0.652·17-s + 0.373·19-s + (−0.491 − 0.0897i)20-s + (0.131 − 0.0756i)22-s + (−0.178 − 0.308i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0931i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 - 0.0931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01964984367\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01964984367\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-3.23 + 3.81i)T \) |
good | 7 | \( 1 + (8.15 + 4.70i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (2.03 + 1.17i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (1.33 - 0.769i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 11.0T + 289T^{2} \) |
| 19 | \( 1 - 7.09T + 361T^{2} \) |
| 23 | \( 1 + (4.09 + 7.09i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (15.5 + 8.97i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-29.3 - 50.8i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 20.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (42.3 - 24.4i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (3.07 + 1.77i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (34.8 - 60.4i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 69.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-34.0 + 19.6i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (57.8 - 100. i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-88.8 + 51.2i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 120. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-9.13 + 15.8i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (80.9 - 140. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 88.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (121. + 70.3i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.03662492933773013623824195247, −9.688120723475049760575280430308, −8.775491627837640431203164999172, −7.997164539272610273309985541341, −6.80031940246291752263172240215, −6.34444174308580995991679796963, −5.24424354325927366462648543070, −4.35003181512380540482331217533, −2.98227685075162957335258702321, −1.34713416638006819787150362540,
0.00742753805797210584940739155, 1.99290428871410632534496941432, 2.83490911330703248857639733778, 3.74072341434878327744755800359, 5.26353149854835480329647891388, 6.22401230263711262129308329607, 6.95747998020376084133874481106, 8.047728851505348198844929623602, 9.187989946955676972181750827753, 9.660994235167603459290584796895