| L(s) = 1 | + (−1.91 − 0.584i)2-s + (2.24 + 1.98i)3-s + (3.31 + 2.23i)4-s + (0.641 + 3.64i)5-s + (−3.13 − 5.11i)6-s + (−2.33 + 2.78i)7-s + (−5.03 − 6.21i)8-s + (1.08 + 8.93i)9-s + (0.899 − 7.33i)10-s + (−15.6 − 2.76i)11-s + (3.00 + 11.6i)12-s + (13.1 + 4.78i)13-s + (6.09 − 3.96i)14-s + (−5.80 + 9.45i)15-s + (6.00 + 14.8i)16-s + (4.98 + 8.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.956 − 0.292i)2-s + (0.748 + 0.663i)3-s + (0.829 + 0.558i)4-s + (0.128 + 0.728i)5-s + (−0.522 − 0.852i)6-s + (−0.333 + 0.397i)7-s + (−0.629 − 0.776i)8-s + (0.120 + 0.992i)9-s + (0.0899 − 0.733i)10-s + (−1.42 − 0.251i)11-s + (0.250 + 0.968i)12-s + (1.01 + 0.367i)13-s + (0.435 − 0.282i)14-s + (−0.386 + 0.630i)15-s + (0.375 + 0.926i)16-s + (0.293 + 0.507i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.816917 + 0.662414i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.816917 + 0.662414i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.91 + 0.584i)T \) |
| 3 | \( 1 + (-2.24 - 1.98i)T \) |
| good | 5 | \( 1 + (-0.641 - 3.64i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (2.33 - 2.78i)T + (-8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (15.6 + 2.76i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (-13.1 - 4.78i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-4.98 - 8.63i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-14.6 - 8.46i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (0.374 + 0.446i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-16.6 + 6.04i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (18.3 + 21.8i)T + (-166. + 946. i)T^{2} \) |
| 37 | \( 1 + (31.7 + 55.0i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-59.2 - 21.5i)T + (1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-1.70 - 0.300i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-51.7 + 61.6i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 66.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-65.4 + 11.5i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-5.43 - 4.55i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (12.7 - 35.1i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-12.7 + 7.38i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-66.1 + 114. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (24.4 + 67.1i)T + (-4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-8.25 - 22.6i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (29.6 - 51.3i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-13.3 + 75.5i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−13.73976261662859161222261711027, −12.59422631182389534622517977474, −11.02208460313660796606563934209, −10.47546448535319067316524446209, −9.445740053926267141193700837401, −8.418410002052440953803658284274, −7.45818493396999781870918505430, −5.85220198238283464566936416724, −3.55675896363506002617186241136, −2.43905948194785939601121678850,
1.02489927911311185568326817667, 2.91661226189439196986513294958, 5.39021091300331919864256874087, 6.89656580168809937672784815920, 7.87377300312316149028435296824, 8.742845404815420626935585720174, 9.744575379408408148052442651859, 10.87062962433356462282949740589, 12.35581249114580813252277724557, 13.25321633625374570522919488528
