| L(s) = 1 | + (−0.696 + 1.20i)2-s + (1.03 + 1.78i)4-s + (7.63 + 4.41i)7-s − 8.43·8-s + (0.805 + 0.464i)11-s + (21.2 − 12.2i)13-s + (−10.6 + 6.13i)14-s + (1.74 − 3.02i)16-s + 18.3·17-s + 5.58·19-s + (−1.12 + 0.647i)22-s + (11.9 + 20.6i)23-s + 34.0i·26-s + 18.1i·28-s + (−23.7 − 13.7i)29-s + ⋯ |
| L(s) = 1 | + (−0.348 + 0.602i)2-s + (0.257 + 0.446i)4-s + (1.09 + 0.630i)7-s − 1.05·8-s + (0.0732 + 0.0422i)11-s + (1.63 − 0.941i)13-s + (−0.759 + 0.438i)14-s + (0.109 − 0.189i)16-s + 1.07·17-s + 0.294·19-s + (−0.0509 + 0.0294i)22-s + (0.517 + 0.896i)23-s + 1.31i·26-s + 0.649i·28-s + (−0.819 − 0.473i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0424 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0424 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.002241065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.002241065\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (0.696 - 1.20i)T + (-2 - 3.46i)T^{2} \) |
| 7 | \( 1 + (-7.63 - 4.41i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-0.805 - 0.464i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-21.2 + 12.2i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 18.3T + 289T^{2} \) |
| 19 | \( 1 - 5.58T + 361T^{2} \) |
| 23 | \( 1 + (-11.9 - 20.6i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (23.7 + 13.7i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-4.66 - 8.08i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 24.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-6.45 + 3.72i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-30.7 - 17.7i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (0.172 - 0.298i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 81.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (65.9 - 38.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (29.6 - 51.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (70.9 - 40.9i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 37.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 3.49iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (62.0 - 107. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (27.9 - 48.4i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 6.78iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (101. + 58.5i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−10.62809902526985476984612540437, −9.324098423696112750453438837529, −8.582494503515985708042080777045, −7.921493703063062682754795910376, −7.28065552303598075567767931820, −5.87972896332807799657506911603, −5.52172932700419573465729769865, −3.88741254654318165816289319970, −2.86330576234057334057914362779, −1.29945739910404283022288324968,
0.995021296379385239817788360466, 1.75301907222451216969292308989, 3.28145489434474798492646368617, 4.40643781382555502292389060368, 5.58551878475000332149825000587, 6.48053339891746321494878180781, 7.52582382022128121333382674492, 8.553973285344397742044794885893, 9.246922157351100680766975833222, 10.32085502614304242242313638520
