L(s) = 1 | + (−0.406 + 0.913i)2-s + (0.809 − 0.587i)3-s + (−0.669 − 0.743i)4-s + (0.978 − 0.207i)5-s + (0.207 + 0.978i)6-s + (−0.994 − 0.104i)7-s + (0.951 − 0.309i)8-s + (0.309 − 0.951i)9-s + (−0.207 + 0.978i)10-s − i·11-s + (−0.978 − 0.207i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (0.669 − 0.743i)15-s + (−0.104 + 0.994i)16-s + (0.743 − 0.669i)17-s + ⋯ |
L(s) = 1 | + (−0.406 + 0.913i)2-s + (0.809 − 0.587i)3-s + (−0.669 − 0.743i)4-s + (0.978 − 0.207i)5-s + (0.207 + 0.978i)6-s + (−0.994 − 0.104i)7-s + (0.951 − 0.309i)8-s + (0.309 − 0.951i)9-s + (−0.207 + 0.978i)10-s − i·11-s + (−0.978 − 0.207i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (0.669 − 0.743i)15-s + (−0.104 + 0.994i)16-s + (0.743 − 0.669i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.553568579 - 0.4383789294i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.553568579 - 0.4383789294i\) |
\(L(1)\) |
\(\approx\) |
\(1.168093842 - 0.03869793043i\) |
\(L(1)\) |
\(\approx\) |
\(1.168093842 - 0.03869793043i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 61 | \( 1 \) |
good | 2 | \( 1 + (-0.406 + 0.913i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.978 - 0.207i)T \) |
| 7 | \( 1 + (-0.994 - 0.104i)T \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (0.743 - 0.669i)T \) |
| 19 | \( 1 + (0.104 + 0.994i)T \) |
| 23 | \( 1 + (0.951 + 0.309i)T \) |
| 29 | \( 1 + (0.866 + 0.5i)T \) |
| 31 | \( 1 + (-0.406 - 0.913i)T \) |
| 37 | \( 1 + (-0.587 + 0.809i)T \) |
| 41 | \( 1 + (0.809 + 0.587i)T \) |
| 43 | \( 1 + (-0.743 - 0.669i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.951 + 0.309i)T \) |
| 59 | \( 1 + (0.406 - 0.913i)T \) |
| 67 | \( 1 + (0.207 + 0.978i)T \) |
| 71 | \( 1 + (-0.207 + 0.978i)T \) |
| 73 | \( 1 + (-0.978 - 0.207i)T \) |
| 79 | \( 1 + (0.743 + 0.669i)T \) |
| 83 | \( 1 + (0.913 + 0.406i)T \) |
| 89 | \( 1 + (0.587 + 0.809i)T \) |
| 97 | \( 1 + (-0.913 + 0.406i)T \) |
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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
−32.275027994122609050660882462481, −31.11403176109435171890224182513, −30.15014511225175652005985060743, −28.90862318206431413973857726210, −28.14546555822826886480261893390, −26.61687295324147269393766812338, −25.94300229583872365561178177439, −25.13659303739807374770439060349, −22.84619845841862127215311614479, −21.74798735978410394064724476681, −21.10211217284586049867177891036, −19.788092529236933942908998311711, −19.02378443491947703151742833059, −17.56838435798978316547260653623, −16.39618717822100481224642699056, −14.70307351046338554726427450538, −13.51359557192934559047427852593, −12.48040994721285156032769941391, −10.54424752381051717449669404293, −9.68393335568170588751763461453, −8.96109028953077210007965963961, −7.06866792601538805098511863009, −4.774520099798563289474447702493, −3.16442963148636069447097621259, −2.01788389520198598888401512404,
0.971924203687201451721885382660, 3.10421470660587801885403598936, 5.5413127685492987352751343072, 6.648061500081994815058153538961, 7.99973041262547441656282626023, 9.25638768289382775375789129142, 10.1089930186871471963461404542, 12.75577130258191238306884757348, 13.64752969795114733291980844130, 14.59444237284228970148313375817, 16.07965635036583790427192136216, 17.18302748290964212565616320068, 18.46452955082966075395335497768, 19.26456136231218639596176105826, 20.60604924890607133443867789354, 22.205151713515060475243889835317, 23.5001007393978708155622784463, 24.895445866749669367315572326870, 25.210659960924360807718004623292, 26.28772568214988010521647560898, 27.34749737849005089657816859350, 29.16306121984120873018004153057, 29.57877805907529445608742116077, 31.55430952071167923902863609863, 32.25474881501169931311642339483