L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s − 7-s + (0.309 + 0.951i)8-s + (0.809 − 0.587i)11-s + (0.809 + 0.587i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)17-s + (0.309 + 0.951i)19-s + (−0.309 + 0.951i)22-s + (−0.809 + 0.587i)23-s − 26-s + (−0.309 + 0.951i)28-s + (−0.309 + 0.951i)29-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s − 7-s + (0.309 + 0.951i)8-s + (0.809 − 0.587i)11-s + (0.809 + 0.587i)13-s + (0.809 − 0.587i)14-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)17-s + (0.309 + 0.951i)19-s + (−0.309 + 0.951i)22-s + (−0.809 + 0.587i)23-s − 26-s + (−0.309 + 0.951i)28-s + (−0.309 + 0.951i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6186552539 + 0.6588009319i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6186552539 + 0.6588009319i\) |
\(L(1)\) |
\(\approx\) |
\(0.6745711995 + 0.2670816318i\) |
\(L(1)\) |
\(\approx\) |
\(0.6745711995 + 0.2670816318i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (0.809 - 0.587i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
| 17 | \( 1 + (0.309 + 0.951i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
| 23 | \( 1 + (-0.809 + 0.587i)T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (0.309 + 0.951i)T \) |
| 37 | \( 1 + (0.809 + 0.587i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (0.309 - 0.951i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (0.809 + 0.587i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 + (-0.309 - 0.951i)T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (0.309 - 0.951i)T \) |
| 83 | \( 1 + (0.309 + 0.951i)T \) |
| 89 | \( 1 + (0.809 - 0.587i)T \) |
| 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
−30.45421313028494474932501023914, −29.78972909650278053946904323632, −28.51520663157403303218353490160, −27.87672844179880460903969386440, −26.56951873835590387829604047053, −25.66400627936200488025477691999, −24.7744599278026936073990638669, −22.871723897208606614152518445975, −22.11648667142112123431858224126, −20.61578927793656665064395323585, −19.85355271015115075915981507350, −18.739107524374311762802986348833, −17.69209528184444908282838552265, −16.49683436631425155559496634625, −15.51289036467063384165577920791, −13.58594927417852583037112132432, −12.458776819574337630127543410418, −11.32603735525613577757663060260, −9.92747520086264918832243951586, −9.121872837726798025008402633897, −7.58869242656452991122921431304, −6.307497343845292934227362329571, −4.003385295682352927990601534257, −2.61090157321330634432956947173, −0.647907608515370417496603222912,
1.38164771200965176663227890790, 3.640475470701649449903715557477, 5.83894634426235414710903452448, 6.677972549457157845210349042613, 8.25206795760349723209219285563, 9.34511289589187023609734684701, 10.46225495016986009119903782230, 11.87500967273793395671203314554, 13.5805229634776756424778887777, 14.74579422935854980789671760157, 16.17446912566509451624708963149, 16.68359267747745592251805682454, 18.17450471056968502257826957388, 19.17122787788890232044546931487, 19.99354281581144988121257733504, 21.607725978389323125843301682163, 22.98045151066318833276914889148, 23.96010920999041099860235042911, 25.21457389958965754837935692096, 25.958008911049505381577090183753, 27.007759824986610116150256778486, 28.120268693628343308049274426298, 29.02790695945098416389827673486, 30.077046485710029839327527119767, 31.75349440960455470478102006366