| L(s) = 1 | + 11.3i·2-s + (63 + 50.9i)3-s − 128.·4-s + (−576 + 712. i)6-s − 2.78e3·7-s − 1.44e3i·8-s + (1.37e3 + 6.41e3i)9-s − 2.24e4i·11-s + (−8.06e3 − 6.51e3i)12-s + 1.31e4·13-s − 3.15e4i·14-s + 1.63e4·16-s − 6.63e4i·17-s + (−7.25e4 + 1.55e4i)18-s + 1.44e5·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.777 + 0.628i)3-s − 0.500·4-s + (−0.444 + 0.549i)6-s − 1.16·7-s − 0.353i·8-s + (0.209 + 0.977i)9-s − 1.53i·11-s + (−0.388 − 0.314i)12-s + 0.460·13-s − 0.820i·14-s + 0.250·16-s − 0.794i·17-s + (−0.691 + 0.148i)18-s + 1.10·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.285720736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.285720736\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 11.3iT \) |
| 3 | \( 1 + (-63 - 50.9i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 2.78e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.24e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 1.31e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 6.63e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.44e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 4.93e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 6.27e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 7.28e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.96e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 9.86e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 7.81e4T + 1.16e13T^{2} \) |
| 47 | \( 1 + 3.51e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 5.22e5iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 5.00e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.75e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.71e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 2.58e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 2.81e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 9.18e6T + 1.51e15T^{2} \) |
| 83 | \( 1 + 8.71e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 8.12e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.28e8T + 7.83e15T^{2} \) |
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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
−11.53743277501105582259268732184, −10.29576854250929208507322711121, −9.353561758236499880391785374726, −8.644036183580951269145022945777, −7.52405346831349193133560820698, −6.27893217666250032158944546792, −5.18596835915265506036388247847, −3.65132127768030371284949660359, −2.95570568720998965527478642815, −0.69864116567979844421179783739,
0.890881242050613215461181789378, 2.13638847736000622670610088380, 3.18961206462748559445358013162, 4.28637595792930959390177693654, 6.12980823260756400173595365979, 7.21347795762443551996285105774, 8.325168506554680118484881811181, 9.616287651803741478186684391059, 9.928621280549931478178685174355, 11.57531460519255250300459718526
