| L(s) = 1 | + (26.4 + 26.4i)2-s + (−94.0 − 227. i)3-s + 884. i·4-s + (−1.43e3 + 592. i)5-s + (3.51e3 − 8.48e3i)6-s + (−8.66e3 − 3.58e3i)7-s + (−9.84e3 + 9.84e3i)8-s + (−2.87e4 + 2.87e4i)9-s + (−5.34e4 − 2.21e4i)10-s + (1.21e4 − 2.94e4i)11-s + (2.00e5 − 8.31e4i)12-s + 3.83e3i·13-s + (−1.34e5 − 3.23e5i)14-s + (2.69e5 + 2.69e5i)15-s − 6.75e4·16-s + (2.71e5 + 2.12e5i)17-s + ⋯ |
| L(s) = 1 | + (1.16 + 1.16i)2-s + (−0.670 − 1.61i)3-s + 1.72i·4-s + (−1.02 + 0.424i)5-s + (1.10 − 2.67i)6-s + (−1.36 − 0.565i)7-s + (−0.850 + 0.850i)8-s + (−1.46 + 1.46i)9-s + (−1.69 − 0.700i)10-s + (0.250 − 0.605i)11-s + (2.79 − 1.15i)12-s + 0.0372i·13-s + (−0.933 − 2.25i)14-s + (1.37 + 1.37i)15-s − 0.257·16-s + (0.787 + 0.616i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.171168 - 0.388315i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.171168 - 0.388315i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (-2.71e5 - 2.12e5i)T \) |
| good | 2 | \( 1 + (-26.4 - 26.4i)T + 512iT^{2} \) |
| 3 | \( 1 + (94.0 + 227. i)T + (-1.39e4 + 1.39e4i)T^{2} \) |
| 5 | \( 1 + (1.43e3 - 592. i)T + (1.38e6 - 1.38e6i)T^{2} \) |
| 7 | \( 1 + (8.66e3 + 3.58e3i)T + (2.85e7 + 2.85e7i)T^{2} \) |
| 11 | \( 1 + (-1.21e4 + 2.94e4i)T + (-1.66e9 - 1.66e9i)T^{2} \) |
| 13 | \( 1 - 3.83e3iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (4.98e5 + 4.98e5i)T + 3.22e11iT^{2} \) |
| 23 | \( 1 + (5.41e4 - 1.30e5i)T + (-1.27e12 - 1.27e12i)T^{2} \) |
| 29 | \( 1 + (4.99e6 - 2.06e6i)T + (1.02e13 - 1.02e13i)T^{2} \) |
| 31 | \( 1 + (3.36e6 + 8.11e6i)T + (-1.86e13 + 1.86e13i)T^{2} \) |
| 37 | \( 1 + (1.12e5 + 2.72e5i)T + (-9.18e13 + 9.18e13i)T^{2} \) |
| 41 | \( 1 + (-1.30e7 - 5.41e6i)T + (2.31e14 + 2.31e14i)T^{2} \) |
| 43 | \( 1 + (1.32e7 - 1.32e7i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + 2.19e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (5.89e7 + 5.89e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + (-7.42e7 + 7.42e7i)T - 8.66e15iT^{2} \) |
| 61 | \( 1 + (9.05e6 + 3.74e6i)T + (8.26e15 + 8.26e15i)T^{2} \) |
| 67 | \( 1 + 1.49e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (2.68e6 + 6.48e6i)T + (-3.24e16 + 3.24e16i)T^{2} \) |
| 73 | \( 1 + (8.42e7 - 3.48e7i)T + (4.16e16 - 4.16e16i)T^{2} \) |
| 79 | \( 1 + (-3.15e7 + 7.62e7i)T + (-8.47e16 - 8.47e16i)T^{2} \) |
| 83 | \( 1 + (4.05e7 + 4.05e7i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 + 6.66e7iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (-5.30e8 + 2.19e8i)T + (5.37e17 - 5.37e17i)T^{2} \) |
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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
−16.29645998504351441499304469480, −14.78777524754386103352965649028, −13.29217498058751831575861063857, −12.76170239028007048070456622944, −11.36087401953563458242698852527, −7.79163668406307489995869725431, −6.88476073597915020847468691372, −5.98281468981082322965021934786, −3.57860480226059371069634356413, −0.15160121659019066136219135735,
3.35326740157492034597163610296, 4.29377954590178175587904974363, 5.68302066363696449780879509371, 9.422918657888268420953058771613, 10.52837947391921475588414520443, 11.88430439743647640478596351607, 12.55374893443063908592097803884, 14.69786789529041902422292912109, 15.72570599084165007619568078414, 16.62447269928330316855827527865
