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
−32.25474881501169931311642339483, −31.55430952071167923902863609863, −29.57877805907529445608742116077, −29.16306121984120873018004153057, −27.34749737849005089657816859350, −26.28772568214988010521647560898, −25.210659960924360807718004623292, −24.895445866749669367315572326870, −23.5001007393978708155622784463, −22.205151713515060475243889835317, −20.60604924890607133443867789354, −19.26456136231218639596176105826, −18.46452955082966075395335497768, −17.18302748290964212565616320068, −16.07965635036583790427192136216, −14.59444237284228970148313375817, −13.64752969795114733291980844130, −12.75577130258191238306884757348, −10.1089930186871471963461404542, −9.25638768289382775375789129142, −7.99973041262547441656282626023, −6.648061500081994815058153538961, −5.5413127685492987352751343072, −3.10421470660587801885403598936, −0.971924203687201451721885382660,
2.01788389520198598888401512404, 3.16442963148636069447097621259, 4.774520099798563289474447702493, 7.06866792601538805098511863009, 8.96109028953077210007965963961, 9.68393335568170588751763461453, 10.54424752381051717449669404293, 12.48040994721285156032769941391, 13.51359557192934559047427852593, 14.70307351046338554726427450538, 16.39618717822100481224642699056, 17.56838435798978316547260653623, 19.02378443491947703151742833059, 19.788092529236933942908998311711, 21.10211217284586049867177891036, 21.74798735978410394064724476681, 22.84619845841862127215311614479, 25.13659303739807374770439060349, 25.94300229583872365561178177439, 26.61687295324147269393766812338, 28.14546555822826886480261893390, 28.90862318206431413973857726210, 30.15014511225175652005985060743, 31.11403176109435171890224182513, 32.275027994122609050660882462481