| L(s) = 1 | + (4.15 + 4.15i)2-s + (101. + 244. i)3-s − 477. i·4-s + (2.36e3 − 977. i)5-s + (−596. + 1.44e3i)6-s + (3.58e3 + 1.48e3i)7-s + (4.11e3 − 4.11e3i)8-s + (−3.57e4 + 3.57e4i)9-s + (1.38e4 + 5.75e3i)10-s + (−8.98e3 + 2.16e4i)11-s + (1.16e5 − 4.84e4i)12-s − 2.43e4i·13-s + (8.72e3 + 2.10e4i)14-s + (4.79e5 + 4.79e5i)15-s − 2.10e5·16-s + (−2.97e5 − 1.72e5i)17-s + ⋯ |
| L(s) = 1 | + (0.183 + 0.183i)2-s + (0.723 + 1.74i)3-s − 0.932i·4-s + (1.68 − 0.699i)5-s + (−0.187 + 0.453i)6-s + (0.564 + 0.233i)7-s + (0.355 − 0.355i)8-s + (−1.81 + 1.81i)9-s + (0.439 + 0.181i)10-s + (−0.185 + 0.446i)11-s + (1.62 − 0.674i)12-s − 0.236i·13-s + (0.0607 + 0.146i)14-s + (2.44 + 2.44i)15-s − 0.801·16-s + (−0.865 − 0.501i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.57434 + 1.35586i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.57434 + 1.35586i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (2.97e5 + 1.72e5i)T \) |
| good | 2 | \( 1 + (-4.15 - 4.15i)T + 512iT^{2} \) |
| 3 | \( 1 + (-101. - 244. i)T + (-1.39e4 + 1.39e4i)T^{2} \) |
| 5 | \( 1 + (-2.36e3 + 977. i)T + (1.38e6 - 1.38e6i)T^{2} \) |
| 7 | \( 1 + (-3.58e3 - 1.48e3i)T + (2.85e7 + 2.85e7i)T^{2} \) |
| 11 | \( 1 + (8.98e3 - 2.16e4i)T + (-1.66e9 - 1.66e9i)T^{2} \) |
| 13 | \( 1 + 2.43e4iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (-2.15e5 - 2.15e5i)T + 3.22e11iT^{2} \) |
| 23 | \( 1 + (1.50e4 - 3.63e4i)T + (-1.27e12 - 1.27e12i)T^{2} \) |
| 29 | \( 1 + (2.91e6 - 1.20e6i)T + (1.02e13 - 1.02e13i)T^{2} \) |
| 31 | \( 1 + (-6.30e5 - 1.52e6i)T + (-1.86e13 + 1.86e13i)T^{2} \) |
| 37 | \( 1 + (1.55e6 + 3.76e6i)T + (-9.18e13 + 9.18e13i)T^{2} \) |
| 41 | \( 1 + (-6.01e6 - 2.49e6i)T + (2.31e14 + 2.31e14i)T^{2} \) |
| 43 | \( 1 + (-9.76e6 + 9.76e6i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + 3.32e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (-2.56e7 - 2.56e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + (3.68e7 - 3.68e7i)T - 8.66e15iT^{2} \) |
| 61 | \( 1 + (1.18e8 + 4.89e7i)T + (8.26e15 + 8.26e15i)T^{2} \) |
| 67 | \( 1 - 1.60e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (1.23e8 + 2.97e8i)T + (-3.24e16 + 3.24e16i)T^{2} \) |
| 73 | \( 1 + (1.92e8 - 7.97e7i)T + (4.16e16 - 4.16e16i)T^{2} \) |
| 79 | \( 1 + (1.02e8 - 2.46e8i)T + (-8.47e16 - 8.47e16i)T^{2} \) |
| 83 | \( 1 + (-3.64e8 - 3.64e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 - 4.25e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (-2.01e8 + 8.35e7i)T + (5.37e17 - 5.37e17i)T^{2} \) |
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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
−16.61712564162678864669663164872, −15.43064031809538985308850477777, −14.40008736082145444521905123208, −13.55254081316147209407012965153, −10.69740041772664270050176022764, −9.725395182442054052866609645429, −8.941543491097840959152614361515, −5.56035995331935961071150090959, −4.75153255695088022855760676040, −2.09383914748976873877270395616,
1.75593222399111276223526015530, 2.78706952053227617541510769014, 6.27088358684339810647446804043, 7.52120267962732709237307139227, 8.948546514928361926273212571940, 11.28333270058078014665412044878, 12.95075134406310008560192362428, 13.58202618006406440512680878733, 14.41973621030938214066416337929, 17.31227415047514727564544091805
