L(s) = 1 | − 3.18i·2-s + (2.94 + 0.581i)3-s − 6.16·4-s + 2.91i·5-s + (1.85 − 9.38i)6-s + 12.1·7-s + 6.88i·8-s + (8.32 + 3.42i)9-s + 9.30·10-s − 13.3i·11-s + (−18.1 − 3.58i)12-s − 12.2·13-s − 38.8i·14-s + (−1.69 + 8.58i)15-s − 2.68·16-s − 2.10i·17-s + ⋯ |
L(s) = 1 | − 1.59i·2-s + (0.981 + 0.193i)3-s − 1.54·4-s + 0.583i·5-s + (0.308 − 1.56i)6-s + 1.74·7-s + 0.860i·8-s + (0.924 + 0.380i)9-s + 0.930·10-s − 1.21i·11-s + (−1.51 − 0.298i)12-s − 0.939·13-s − 2.77i·14-s + (−0.113 + 0.572i)15-s − 0.168·16-s − 0.123i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.193 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.37006 - 1.66730i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37006 - 1.66730i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.94 - 0.581i)T \) |
| 59 | \( 1 + 7.68iT \) |
good | 2 | \( 1 + 3.18iT - 4T^{2} \) |
| 5 | \( 1 - 2.91iT - 25T^{2} \) |
| 7 | \( 1 - 12.1T + 49T^{2} \) |
| 11 | \( 1 + 13.3iT - 121T^{2} \) |
| 13 | \( 1 + 12.2T + 169T^{2} \) |
| 17 | \( 1 + 2.10iT - 289T^{2} \) |
| 19 | \( 1 + 3.44T + 361T^{2} \) |
| 23 | \( 1 - 44.1iT - 529T^{2} \) |
| 29 | \( 1 + 41.6iT - 841T^{2} \) |
| 31 | \( 1 + 28.5T + 961T^{2} \) |
| 37 | \( 1 + 16.7T + 1.36e3T^{2} \) |
| 41 | \( 1 - 7.60iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 46.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 1.11iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 30.2iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 46.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 66.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 99.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 129.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 12.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 41.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 75.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 186.T + 9.40e3T^{2} \) |
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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
−11.80986452603418945321381866393, −11.22094550863131324466549622685, −10.40223776463081203267378506159, −9.378338640458043638413994611681, −8.380650229258332160458514468024, −7.41383278075481547947907192918, −5.11945559805448807799653509879, −3.88976825281058540464307162923, −2.74808253030598608378362665091, −1.58792567305277514726525812931,
1.95365789034734460745301789476, 4.59220288623545738047359064372, 4.96121630842498373676780581884, 6.87375994108300006168271203997, 7.58211842909022703833304475567, 8.454256313015338807275187187872, 9.035018622978788462321036831291, 10.46499692290273242172404974027, 12.16908288200946975508170798425, 12.98430906768865055964207625716