L(s) = 1 | + 4.61i·2-s + (−3.98 + 3.33i)3-s − 13.2·4-s − 5·5-s + (−15.3 − 18.3i)6-s + (0.582 − 18.5i)7-s − 24.3i·8-s + (4.79 − 26.5i)9-s − 23.0i·10-s + 65.3i·11-s + (52.9 − 44.2i)12-s − 30.9i·13-s + (85.4 + 2.68i)14-s + (19.9 − 16.6i)15-s + 6.26·16-s − 85.8·17-s + ⋯ |
L(s) = 1 | + 1.63i·2-s + (−0.767 + 0.641i)3-s − 1.66·4-s − 0.447·5-s + (−1.04 − 1.25i)6-s + (0.0314 − 0.999i)7-s − 1.07i·8-s + (0.177 − 0.984i)9-s − 0.729i·10-s + 1.79i·11-s + (1.27 − 1.06i)12-s − 0.660i·13-s + (1.63 + 0.0513i)14-s + (0.343 − 0.286i)15-s + 0.0978·16-s − 1.22·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0786760 - 0.0382992i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0786760 - 0.0382992i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (3.98 - 3.33i)T \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 + (-0.582 + 18.5i)T \) |
good | 2 | \( 1 - 4.61iT - 8T^{2} \) |
| 11 | \( 1 - 65.3iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 30.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 85.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 6.24iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 193. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 30.7iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 61.0iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 8.71T + 5.06e4T^{2} \) |
| 41 | \( 1 + 387.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 208.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 211. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 213.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 673. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 500.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 319. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 555. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 842.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 887.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 480.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 477. iT - 9.12e5T^{2} \) |
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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
−14.80680317288135634656262555501, −13.35118868907728582384063343864, −12.23787078229924554516646562317, −10.76683700178118052330973375250, −9.839443608633965039503486447464, −8.462056876984018956211250983694, −7.16420084232880479964881169874, −6.55527728052814139113362275397, −4.89663307006279600320388717710, −4.29448337803343816912534659910,
0.05316000204138030792666177034, 1.78715047372656069361369965753, 3.34670194319871241613295899271, 5.06351775123610793475229695894, 6.44856488002341733884559976079, 8.296501963354670252692139224179, 9.294809705625033537730978689636, 10.85415676764904130514891679097, 11.50413990718433685651694436208, 11.92417198881976005882289394774