L(s) = 1 | + (−0.587 − 0.809i)3-s + (0.951 − 0.309i)5-s + (0.809 + 0.587i)7-s + (−0.309 + 0.951i)9-s + (0.951 + 0.309i)13-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)17-s + (−0.587 − 0.809i)19-s − i·21-s − 23-s + (0.809 − 0.587i)25-s + (0.951 − 0.309i)27-s + (0.587 − 0.809i)29-s + (0.309 − 0.951i)31-s + (0.951 + 0.309i)35-s + ⋯ |
L(s) = 1 | + (−0.587 − 0.809i)3-s + (0.951 − 0.309i)5-s + (0.809 + 0.587i)7-s + (−0.309 + 0.951i)9-s + (0.951 + 0.309i)13-s + (−0.809 − 0.587i)15-s + (0.309 + 0.951i)17-s + (−0.587 − 0.809i)19-s − i·21-s − 23-s + (0.809 − 0.587i)25-s + (0.951 − 0.309i)27-s + (0.587 − 0.809i)29-s + (0.309 − 0.951i)31-s + (0.951 + 0.309i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.151359617 - 0.3395421650i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.151359617 - 0.3395421650i\) |
\(L(1)\) |
\(\approx\) |
\(1.084269641 - 0.2229968010i\) |
\(L(1)\) |
\(\approx\) |
\(1.084269641 - 0.2229968010i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (-0.587 - 0.809i)T \) |
| 5 | \( 1 + (0.951 - 0.309i)T \) |
| 7 | \( 1 + (0.809 + 0.587i)T \) |
| 13 | \( 1 + (0.951 + 0.309i)T \) |
| 17 | \( 1 + (0.309 + 0.951i)T \) |
| 19 | \( 1 + (-0.587 - 0.809i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.587 - 0.809i)T \) |
| 31 | \( 1 + (0.309 - 0.951i)T \) |
| 37 | \( 1 + (-0.587 + 0.809i)T \) |
| 41 | \( 1 + (0.809 - 0.587i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.809 + 0.587i)T \) |
| 53 | \( 1 + (-0.951 - 0.309i)T \) |
| 59 | \( 1 + (-0.587 + 0.809i)T \) |
| 61 | \( 1 + (-0.951 + 0.309i)T \) |
| 67 | \( 1 + iT \) |
| 71 | \( 1 + (-0.309 - 0.951i)T \) |
| 73 | \( 1 + (0.809 + 0.587i)T \) |
| 79 | \( 1 + (0.309 - 0.951i)T \) |
| 83 | \( 1 + (-0.951 + 0.309i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (0.309 - 0.951i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.53766116384214648278770491828, −26.63481160141652424086714364427, −25.735337140789610488378139919764, −24.703206521472242390919196138850, −23.3557455506331838311502066103, −22.781378210222444238156059614612, −21.492320446833927255726090984464, −21.043902835908273261851905379031, −20.09979828793710112954424928421, −18.24927865911735701069932371454, −17.80748426119096913165566436473, −16.73674798406161690496259568202, −15.875966365327498919323041476098, −14.49602884570148604196307769126, −13.93282213756496101886290608733, −12.43958584494704134565795985785, −11.11525109557949386873667938864, −10.48510832949185301551499363211, −9.525511020457568469191294724515, −8.225695017695942845013601598725, −6.64084139726066347507210367907, −5.638905930232108566269704713717, −4.59922751846235182436760029962, −3.27212816759386222715352038933, −1.440029303350501005614341196252,
1.40812656019102209928066007356, 2.32072053764644670471483501257, 4.533016058229276416312892522881, 5.77825504122755555893440368289, 6.36649040833402979168270842281, 7.98095112394808131512495663725, 8.833822441347579555780186001670, 10.348505479521055386432365643441, 11.38309684347593311298478558890, 12.38124275024979873227964002965, 13.38107902475614448501657715952, 14.18081136126682837273637208607, 15.573276933928883731097155303100, 16.91765789223704557225602705576, 17.59403802620364570639247956183, 18.37120972538441019891119690228, 19.326211060996644877526739899937, 20.7831840498015471612109365440, 21.54696486537547593449692207487, 22.47997192748235887314778667869, 23.85241093583886843720719423597, 24.24604437553900076791816079446, 25.36233368121133173777389335120, 26.03223218163391975764473098355, 27.87876868826721167342094756663