L(s) = 1 | + (−3.70e3 − 6.42e3i)5-s + (−4.08e4 + 1.75e4i)7-s + (5.92e5 + 3.42e5i)11-s + 1.28e6i·13-s + (−2.64e6 + 4.57e6i)17-s + (−1.55e7 + 8.96e6i)19-s + (3.79e6 − 2.19e6i)23-s + (−3.09e6 + 5.35e6i)25-s − 6.56e7i·29-s + (−1.57e8 − 9.09e7i)31-s + (2.64e8 + 1.97e8i)35-s + (1.92e8 + 3.34e8i)37-s + 1.03e9·41-s + 9.00e8·43-s + (1.08e9 + 1.87e9i)47-s + ⋯ |
L(s) = 1 | + (−0.530 − 0.919i)5-s + (−0.918 + 0.395i)7-s + (1.11 + 0.640i)11-s + 0.958i·13-s + (−0.451 + 0.781i)17-s + (−1.43 + 0.830i)19-s + (0.122 − 0.0709i)23-s + (−0.0633 + 0.109i)25-s − 0.594i·29-s + (−0.988 − 0.570i)31-s + (0.851 + 0.634i)35-s + (0.457 + 0.791i)37-s + 1.39·41-s + 0.934·43-s + (0.686 + 1.18i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.08616287105\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.08616287105\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (4.08e4 - 1.75e4i)T \) |
good | 5 | \( 1 + (3.70e3 + 6.42e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (-5.92e5 - 3.42e5i)T + (1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 - 1.28e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + (2.64e6 - 4.57e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (1.55e7 - 8.96e6i)T + (5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-3.79e6 + 2.19e6i)T + (4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 6.56e7iT - 1.22e16T^{2} \) |
| 31 | \( 1 + (1.57e8 + 9.09e7i)T + (1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-1.92e8 - 3.34e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 - 1.03e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 9.00e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-1.08e9 - 1.87e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-1.80e8 - 1.04e8i)T + (4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (1.71e9 - 2.96e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (2.95e9 - 1.70e9i)T + (2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (-4.27e9 + 7.40e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 + 2.58e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 + (5.59e9 + 3.22e9i)T + (1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-6.48e9 - 1.12e10i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 1.06e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (3.57e10 + 6.19e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 - 5.47e10iT - 7.15e21T^{2} \) |
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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
−9.379172251036600353062759565808, −8.995841009312764544584130434658, −7.907773068311845047411842185533, −6.62705560431480821501413328305, −5.96296092485728214849238048773, −4.25445488754747562837306476136, −4.08349744811418687490366388641, −2.34439925190218817405789472920, −1.31028172758346786188334761450, −0.02064157853882960749646336936,
0.805301597375203283638349302426, 2.53121273324024351079357680077, 3.38234153008783312028358338070, 4.20100160170730352255301868837, 5.77143200741665333844634023827, 6.77845424786999700880058124255, 7.28950470805578490680290035787, 8.686917002796080341539948679337, 9.484014677031892883092583996756, 10.87843893363038517274507886901