| L(s) = 1 | + (0.984 − 0.173i)5-s + (0.5 + 0.866i)7-s − i·11-s + (−0.984 − 0.173i)13-s + (−0.173 − 0.984i)17-s + (0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.642 + 0.766i)29-s + 31-s + (0.642 + 0.766i)35-s + i·37-s + (−0.939 − 0.342i)41-s + (0.642 + 0.766i)43-s + (−0.766 − 0.642i)47-s + (−0.5 + 0.866i)49-s + ⋯ |
| L(s) = 1 | + (0.984 − 0.173i)5-s + (0.5 + 0.866i)7-s − i·11-s + (−0.984 − 0.173i)13-s + (−0.173 − 0.984i)17-s + (0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.642 + 0.766i)29-s + 31-s + (0.642 + 0.766i)35-s + i·37-s + (−0.939 − 0.342i)41-s + (0.642 + 0.766i)43-s + (−0.766 − 0.642i)47-s + (−0.5 + 0.866i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.524 + 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.215152367 + 1.237200170i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.215152367 + 1.237200170i\) |
| \(L(1)\) |
\(\approx\) |
\(1.278894132 + 0.09079505921i\) |
| \(L(1)\) |
\(\approx\) |
\(1.278894132 + 0.09079505921i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 + (0.984 - 0.173i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-0.984 - 0.173i)T \) |
| 17 | \( 1 + (-0.173 - 0.984i)T \) |
| 23 | \( 1 + (0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.642 + 0.766i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.939 - 0.342i)T \) |
| 43 | \( 1 + (0.642 + 0.766i)T \) |
| 47 | \( 1 + (-0.766 - 0.642i)T \) |
| 53 | \( 1 + (0.342 + 0.939i)T \) |
| 59 | \( 1 + (-0.642 - 0.766i)T \) |
| 61 | \( 1 + (-0.984 - 0.173i)T \) |
| 67 | \( 1 + (0.342 + 0.939i)T \) |
| 71 | \( 1 + (-0.939 - 0.342i)T \) |
| 73 | \( 1 + (-0.766 + 0.642i)T \) |
| 79 | \( 1 + (0.173 + 0.984i)T \) |
| 83 | \( 1 + (0.866 - 0.5i)T \) |
| 89 | \( 1 + (0.766 + 0.642i)T \) |
| 97 | \( 1 + (-0.939 - 0.342i)T \) |
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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
−19.03060205986278997639224004, −18.04541550098994125556150455787, −17.40895394667098539885624559806, −17.13822899022734815946605707545, −16.39848435466896740679223561630, −15.00289976373059505239946410866, −14.88903954114982348594980781612, −14.00746699616092583532832613312, −13.32428113372448979656988039593, −12.66721610291521474132739897629, −11.88983325277940997943342581011, −10.84832384920541639556053928669, −10.336400838252669325044623013685, −9.72623000183155593944955839981, −8.97151164037520451395800041250, −7.95465632399757855336895749125, −7.21996316748658092298243592466, −6.61463705959045522707971534096, −5.73040891069871702698878922478, −4.73846233511056597356282002260, −4.35700133599165345954044665483, −3.1329401609780114657581409404, −2.1068120052328766128003905738, −1.641054252906583261378159063, −0.435120720117016685015367163106,
0.85895538758196104549144809878, 1.740558846629313392100404853, 2.68089823086305857947749636914, 3.16796214593826318932452476641, 4.73557036683883602910780320608, 5.155121314216122659605798977119, 5.85485222082616127465704311307, 6.66282655474081746833260351140, 7.574964852779774358553033656326, 8.49348128911096372911350821153, 9.11345086306280003397073453814, 9.69802314072002419542413777979, 10.569625458980875331257694850685, 11.422838226554247860657670744713, 11.98904537152996451179351560696, 12.87852575247023965281497377071, 13.57242806030160713003513805260, 14.18914218599058449900100805911, 14.91695820819225197591960295756, 15.6209013468338249899505656626, 16.5147340284240285796614626359, 17.13988902608954577844715398071, 17.76759045600246428910794876035, 18.520209720371572615136000370580, 18.9921699516178346009150631933
