L(s) = 1 | + (−0.841 − 2.03i)2-s + (0.0897 + 0.134i)3-s + (−0.590 + 0.590i)4-s + (−1.04 + 5.26i)5-s + (0.197 − 0.295i)6-s + (1.21 + 6.12i)7-s + (−6.42 − 2.66i)8-s + (3.43 − 8.29i)9-s + (11.5 − 2.30i)10-s + (−12.1 − 8.11i)11-s + (−0.132 − 0.0263i)12-s + (4.79 + 4.79i)13-s + (11.4 − 7.62i)14-s + (−0.801 + 0.331i)15-s + 18.6i·16-s + (6.50 + 15.7i)17-s + ⋯ |
L(s) = 1 | + (−0.420 − 1.01i)2-s + (0.0299 + 0.0447i)3-s + (−0.147 + 0.147i)4-s + (−0.209 + 1.05i)5-s + (0.0329 − 0.0492i)6-s + (0.174 + 0.874i)7-s + (−0.803 − 0.332i)8-s + (0.381 − 0.921i)9-s + (1.15 − 0.230i)10-s + (−1.10 − 0.737i)11-s + (−0.0110 − 0.00219i)12-s + (0.369 + 0.369i)13-s + (0.815 − 0.544i)14-s + (−0.0534 + 0.0221i)15-s + 1.16i·16-s + (0.382 + 0.923i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.618160 - 0.317877i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.618160 - 0.317877i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (-6.50 - 15.7i)T \) |
good | 2 | \( 1 + (0.841 + 2.03i)T + (-2.82 + 2.82i)T^{2} \) |
| 3 | \( 1 + (-0.0897 - 0.134i)T + (-3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (1.04 - 5.26i)T + (-23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (-1.21 - 6.12i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (12.1 + 8.11i)T + (46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-4.79 - 4.79i)T + 169iT^{2} \) |
| 19 | \( 1 + (9.56 + 23.0i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-7.27 + 10.8i)T + (-202. - 488. i)T^{2} \) |
| 29 | \( 1 + (-32.3 - 6.44i)T + (776. + 321. i)T^{2} \) |
| 31 | \( 1 + (1.00 - 0.674i)T + (367. - 887. i)T^{2} \) |
| 37 | \( 1 + (25.8 + 38.6i)T + (-523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-6.13 - 30.8i)T + (-1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (27.8 - 67.3i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (10.4 + 10.4i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (1.97 + 4.77i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-26.1 - 10.8i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (81.0 - 16.1i)T + (3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 + 44.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-32.1 - 48.1i)T + (-1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (0.262 - 1.32i)T + (-4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (24.7 + 16.5i)T + (2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (62.2 - 25.7i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-90.1 + 90.1i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (66.3 + 13.1i)T + (8.69e3 + 3.60e3i)T^{2} \) |
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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
−18.60753229988716760539972469048, −18.03206053020274594876635796153, −15.66495948809099481420889097149, −14.80777743892533505890195500016, −12.72674105458849979929443727370, −11.36300872652885777138707335276, −10.40835665577570428983083355108, −8.779639353973859345825606974740, −6.39279628524038001392938270385, −2.95916334263223198660524035431,
5.06351843953026264270071707092, 7.39486115882744682103638689916, 8.318719143872788158483086939533, 10.28207272462103928356498155407, 12.34539181456178569352563527107, 13.72158070177151006942836165839, 15.53231800887825229265193031966, 16.37901499486708891759320594761, 17.27034132980581125914538385480, 18.61816235830431312333070580805