| L(s) = 1 | + (3.60 + 1.31i)2-s + (8.19 + 6.87i)4-s + (−2.60 − 4.26i)5-s + (5.01 + 5.97i)7-s + (12.8 + 22.2i)8-s + (−3.80 − 18.7i)10-s + (9.63 − 1.69i)11-s + (−3.91 − 10.7i)13-s + (10.2 + 28.0i)14-s + (9.67 + 54.8i)16-s + (−4.13 + 7.16i)17-s + (6.12 + 10.6i)19-s + (7.94 − 52.9i)20-s + (36.9 + 6.51i)22-s + (−15.9 − 13.4i)23-s + ⋯ |
| L(s) = 1 | + (1.80 + 0.655i)2-s + (2.04 + 1.71i)4-s + (−0.521 − 0.852i)5-s + (0.715 + 0.853i)7-s + (1.60 + 2.78i)8-s + (−0.380 − 1.87i)10-s + (0.875 − 0.154i)11-s + (−0.301 − 0.827i)13-s + (0.730 + 2.00i)14-s + (0.604 + 3.42i)16-s + (−0.243 + 0.421i)17-s + (0.322 + 0.558i)19-s + (0.397 − 2.64i)20-s + (1.67 + 0.296i)22-s + (−0.695 − 0.583i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.22530 + 2.58855i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.22530 + 2.58855i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (2.60 + 4.26i)T \) |
| good | 2 | \( 1 + (-3.60 - 1.31i)T + (3.06 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-5.01 - 5.97i)T + (-8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-9.63 + 1.69i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (3.91 + 10.7i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (4.13 - 7.16i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-6.12 - 10.6i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (15.9 + 13.4i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (4.79 - 13.1i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (32.9 + 27.6i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-9.27 - 5.35i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (14.9 + 40.9i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-43.1 + 7.61i)T + (1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-57.2 + 48.0i)T + (383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 2.58T + 2.80e3T^{2} \) |
| 59 | \( 1 + (108. + 19.0i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-12.2 + 10.2i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (10.8 + 29.9i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (65.2 + 37.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-6.13 + 3.53i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (32.2 + 11.7i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (24.1 + 8.77i)T + (5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (66.9 - 38.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (88.6 - 15.6i)T + (8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.78795714950360657565036736620, −10.80962630835929950207833929453, −8.977227387561345429877114439823, −8.111227570125541677148230741707, −7.35318950346090554144707120434, −5.97648139287000698070696545837, −5.41108922300146348443298230452, −4.41618198276564302472343171432, −3.56701955914371986056222130597, −2.02421615538532835077464027532,
1.54529630772105100896751972399, 2.89069171847586711758783110711, 4.07351638177688518682003420629, 4.50854933539244277748542325903, 5.91621078059688224794949157285, 6.96083950124876512197011660532, 7.47339284548755653926338601355, 9.452581787216323419239966363292, 10.54856527742388713272243733798, 11.28093022834182619159769356407
