L(s) = 1 | + (−0.327 + 0.945i)2-s + (0.415 − 0.909i)3-s + (−0.786 − 0.618i)4-s + (−0.959 − 0.281i)5-s + (0.723 + 0.690i)6-s + (0.981 − 0.189i)7-s + (0.841 − 0.540i)8-s + (−0.654 − 0.755i)9-s + (0.580 − 0.814i)10-s + (0.723 − 0.690i)11-s + (−0.888 + 0.458i)12-s + (0.0475 − 0.998i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (0.235 + 0.971i)16-s + (−0.786 + 0.618i)17-s + ⋯ |
L(s) = 1 | + (−0.327 + 0.945i)2-s + (0.415 − 0.909i)3-s + (−0.786 − 0.618i)4-s + (−0.959 − 0.281i)5-s + (0.723 + 0.690i)6-s + (0.981 − 0.189i)7-s + (0.841 − 0.540i)8-s + (−0.654 − 0.755i)9-s + (0.580 − 0.814i)10-s + (0.723 − 0.690i)11-s + (−0.888 + 0.458i)12-s + (0.0475 − 0.998i)13-s + (−0.142 + 0.989i)14-s + (−0.654 + 0.755i)15-s + (0.235 + 0.971i)16-s + (−0.786 + 0.618i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7492964729 - 0.1855687512i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7492964729 - 0.1855687512i\) |
\(L(1)\) |
\(\approx\) |
\(0.8679368596 - 0.04745357172i\) |
\(L(1)\) |
\(\approx\) |
\(0.8679368596 - 0.04745357172i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 67 | \( 1 \) |
good | 2 | \( 1 + (-0.327 + 0.945i)T \) |
| 3 | \( 1 + (0.415 - 0.909i)T \) |
| 5 | \( 1 + (-0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.981 - 0.189i)T \) |
| 11 | \( 1 + (0.723 - 0.690i)T \) |
| 13 | \( 1 + (0.0475 - 0.998i)T \) |
| 17 | \( 1 + (-0.786 + 0.618i)T \) |
| 19 | \( 1 + (0.981 + 0.189i)T \) |
| 23 | \( 1 + (-0.995 + 0.0950i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.0475 + 0.998i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.928 + 0.371i)T \) |
| 43 | \( 1 + (-0.142 - 0.989i)T \) |
| 47 | \( 1 + (0.580 + 0.814i)T \) |
| 53 | \( 1 + (-0.142 + 0.989i)T \) |
| 59 | \( 1 + (0.841 - 0.540i)T \) |
| 61 | \( 1 + (0.723 + 0.690i)T \) |
| 71 | \( 1 + (-0.786 - 0.618i)T \) |
| 73 | \( 1 + (0.723 + 0.690i)T \) |
| 79 | \( 1 + (-0.888 + 0.458i)T \) |
| 83 | \( 1 + (0.235 + 0.971i)T \) |
| 89 | \( 1 + (0.415 + 0.909i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−31.553926166091846326295322124004, −31.08992670347259290615659363115, −30.19978082461648686850430451590, −28.41416688227316714758633361483, −27.739587509479179799422993459317, −26.851165978601509017305126807920, −26.1147300693187139120954056606, −24.373712540302355235373596821761, −22.73025261994922652641989991148, −21.971999294270903012792891492328, −20.67438670421583788399339048724, −20.03335999326080824786445752801, −18.876009220735533232926290405563, −17.57746501570574829763625332973, −16.18939253699611073369642725726, −14.85593208080427502794458010296, −13.84752050474167424910229471070, −11.7221656927878087776155340966, −11.37859362603248039669127172114, −9.81583390770055089072448434894, −8.74382958658421988139989894188, −7.56304947628579913487622443699, −4.674471022641845054633186923835, −3.91391010053088313586512603635, −2.22679579646755516324469845772,
1.16590467472096318942326007714, 3.841669938420712557298148518155, 5.59053572189227000832371528448, 7.162561215364008887270241104565, 8.08532610699734307881781029376, 8.850189117287163057577608172639, 10.99518796060256176299056814761, 12.42098114260366220419814346816, 13.85162479774743859882121880286, 14.74762793533473119096764898239, 15.95114720897401618100692643987, 17.37926303479539347700597782434, 18.22289108831875520451155591975, 19.48575700057573767401990726051, 20.24210935441086887925905114445, 22.35666136593634562160753361710, 23.64103166219974861653850395100, 24.28132924695124794978654434831, 25.00771348670756018758762659672, 26.48847998507363446992575999382, 27.288054611172430946131287434062, 28.35715728845352307740099403615, 30.03435867599763825455200319921, 30.98404049496636086906507140131, 31.91180145287161943687017149310