| L(s) = 1 | + (−10.5 + 10.5i)2-s + (−129. − 53.8i)3-s + 290. i·4-s + (−588. + 1.42e3i)5-s + (1.93e3 − 801. i)6-s + (−818. − 1.97e3i)7-s + (−8.44e3 − 8.44e3i)8-s + (62.9 + 62.9i)9-s + (−8.75e3 − 2.11e4i)10-s + (7.76e4 − 3.21e4i)11-s + (1.56e4 − 3.77e4i)12-s + 1.27e4i·13-s + (2.94e4 + 1.21e4i)14-s + (1.52e5 − 1.52e5i)15-s + 2.91e4·16-s + (−3.19e5 − 1.29e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.465 + 0.465i)2-s + (−0.925 − 0.383i)3-s + 0.567i·4-s + (−0.421 + 1.01i)5-s + (0.609 − 0.252i)6-s + (−0.128 − 0.311i)7-s + (−0.729 − 0.729i)8-s + (0.00319 + 0.00319i)9-s + (−0.277 − 0.668i)10-s + (1.59 − 0.662i)11-s + (0.217 − 0.525i)12-s + 0.123i·13-s + (0.204 + 0.0847i)14-s + (0.779 − 0.779i)15-s + 0.111·16-s + (−0.926 − 0.375i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.431 + 0.901i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.431 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.359498 - 0.226461i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.359498 - 0.226461i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (3.19e5 + 1.29e5i)T \) |
| good | 2 | \( 1 + (10.5 - 10.5i)T - 512iT^{2} \) |
| 3 | \( 1 + (129. + 53.8i)T + (1.39e4 + 1.39e4i)T^{2} \) |
| 5 | \( 1 + (588. - 1.42e3i)T + (-1.38e6 - 1.38e6i)T^{2} \) |
| 7 | \( 1 + (818. + 1.97e3i)T + (-2.85e7 + 2.85e7i)T^{2} \) |
| 11 | \( 1 + (-7.76e4 + 3.21e4i)T + (1.66e9 - 1.66e9i)T^{2} \) |
| 13 | \( 1 - 1.27e4iT - 1.06e10T^{2} \) |
| 19 | \( 1 + (-5.52e5 + 5.52e5i)T - 3.22e11iT^{2} \) |
| 23 | \( 1 + (1.22e6 - 5.07e5i)T + (1.27e12 - 1.27e12i)T^{2} \) |
| 29 | \( 1 + (-8.36e5 + 2.02e6i)T + (-1.02e13 - 1.02e13i)T^{2} \) |
| 31 | \( 1 + (2.69e6 + 1.11e6i)T + (1.86e13 + 1.86e13i)T^{2} \) |
| 37 | \( 1 + (-8.88e5 - 3.68e5i)T + (9.18e13 + 9.18e13i)T^{2} \) |
| 41 | \( 1 + (1.11e7 + 2.70e7i)T + (-2.31e14 + 2.31e14i)T^{2} \) |
| 43 | \( 1 + (6.88e6 + 6.88e6i)T + 5.02e14iT^{2} \) |
| 47 | \( 1 + 1.22e7iT - 1.11e15T^{2} \) |
| 53 | \( 1 + (3.83e7 - 3.83e7i)T - 3.29e15iT^{2} \) |
| 59 | \( 1 + (7.83e7 + 7.83e7i)T + 8.66e15iT^{2} \) |
| 61 | \( 1 + (3.25e7 + 7.85e7i)T + (-8.26e15 + 8.26e15i)T^{2} \) |
| 67 | \( 1 - 1.27e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + (3.23e8 + 1.34e8i)T + (3.24e16 + 3.24e16i)T^{2} \) |
| 73 | \( 1 + (-9.63e7 + 2.32e8i)T + (-4.16e16 - 4.16e16i)T^{2} \) |
| 79 | \( 1 + (-1.51e8 + 6.26e7i)T + (8.47e16 - 8.47e16i)T^{2} \) |
| 83 | \( 1 + (5.56e8 - 5.56e8i)T - 1.86e17iT^{2} \) |
| 89 | \( 1 - 2.83e8iT - 3.50e17T^{2} \) |
| 97 | \( 1 + (1.00e8 - 2.42e8i)T + (-5.37e17 - 5.37e17i)T^{2} \) |
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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
−16.79944074597905768063968180304, −15.51422375814893831158211575268, −13.85743770074721365733747776268, −11.96337688551465015149167008652, −11.24166829825256553925093435568, −9.102782218492851168415562408026, −7.18878999689420573776647653623, −6.40576815821440879435003924901, −3.54234380639799653130910572437, −0.30478956938458136896847454652,
1.34690283795747042415129863722, 4.59000751020119908619265565221, 6.06788170509921199597213060680, 8.716863879779096980114317052192, 9.956448992604218525635577655079, 11.46498684858257147482912389946, 12.26345627011233961553875706856, 14.46176705759826239119385101380, 15.96677938348631571086563525500, 17.01347594789066546873139744265
