L(s) = 1 | − 9.15·2-s + 105. i·3-s − 428.·4-s − 1.75e3i·5-s − 968. i·6-s + 5.86e3i·7-s + 8.60e3·8-s + 8.49e3·9-s + 1.60e4i·10-s − 5.72e4i·11-s − 4.52e4i·12-s + 1.83e5·13-s − 5.37e4i·14-s + 1.85e5·15-s + 1.40e5·16-s + (−3.07e5 − 1.54e5i)17-s + ⋯ |
L(s) = 1 | − 0.404·2-s + 0.753i·3-s − 0.836·4-s − 1.25i·5-s − 0.304i·6-s + 0.924i·7-s + 0.742·8-s + 0.431·9-s + 0.507i·10-s − 1.17i·11-s − 0.630i·12-s + 1.77·13-s − 0.373i·14-s + 0.945·15-s + 0.535·16-s + (−0.894 − 0.448i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.448i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.894 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.13725 - 0.269026i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13725 - 0.269026i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 17 | \( 1 + (3.07e5 + 1.54e5i)T \) |
good | 2 | \( 1 + 9.15T + 512T^{2} \) |
| 3 | \( 1 - 105. iT - 1.96e4T^{2} \) |
| 5 | \( 1 + 1.75e3iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 5.86e3iT - 4.03e7T^{2} \) |
| 11 | \( 1 + 5.72e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.83e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 2.69e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.82e6iT - 1.80e12T^{2} \) |
| 29 | \( 1 + 1.33e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 - 5.82e6iT - 2.64e13T^{2} \) |
| 37 | \( 1 + 1.79e7iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 1.95e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 - 1.23e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.64e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.39e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 6.25e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.82e8iT - 1.16e16T^{2} \) |
| 67 | \( 1 - 1.37e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 5.84e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 + 1.40e8iT - 5.88e16T^{2} \) |
| 79 | \( 1 + 1.26e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 - 9.17e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 5.55e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 6.97e8iT - 7.60e17T^{2} \) |
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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
−16.35855177145799798932475921475, −15.84904527262592743602175406898, −13.81075541166723876262127970651, −12.66959960837938908847981828033, −10.77917989376835224611284276353, −8.924046913279254109433277900104, −8.743897385662560489488728996679, −5.49535396511526509017902092720, −4.09640182754312549871111592979, −0.846309112302970345245977670764,
1.34885839633621012970199457379, 3.97221310213751672417740183761, 6.69175767105858679107889977219, 7.81284594028677730885701790446, 9.808115917721173641712512479219, 11.01430902984881315234953542036, 13.12609091337998486197775275995, 13.84340458556390903534166882833, 15.42340964844686210014994034454, 17.40599294384045414570757205193