L(s) = 1 | + (−0.241 + 0.970i)2-s + (−0.374 + 0.927i)3-s + (−0.882 − 0.469i)4-s + (0.961 + 0.275i)5-s + (−0.809 − 0.587i)6-s + (0.0348 + 0.999i)7-s + (0.669 − 0.743i)8-s + (−0.719 − 0.694i)9-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (0.438 + 0.898i)13-s + (−0.978 − 0.207i)14-s + (−0.615 + 0.788i)15-s + (0.559 + 0.829i)16-s + (0.438 − 0.898i)17-s + (0.848 − 0.529i)18-s + ⋯ |
L(s) = 1 | + (−0.241 + 0.970i)2-s + (−0.374 + 0.927i)3-s + (−0.882 − 0.469i)4-s + (0.961 + 0.275i)5-s + (−0.809 − 0.587i)6-s + (0.0348 + 0.999i)7-s + (0.669 − 0.743i)8-s + (−0.719 − 0.694i)9-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (0.438 + 0.898i)13-s + (−0.978 − 0.207i)14-s + (−0.615 + 0.788i)15-s + (0.559 + 0.829i)16-s + (0.438 − 0.898i)17-s + (0.848 − 0.529i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1334209588 + 0.9451772466i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1334209588 + 0.9451772466i\) |
\(L(1)\) |
\(\approx\) |
\(0.4802857261 + 0.7228558621i\) |
\(L(1)\) |
\(\approx\) |
\(0.4802857261 + 0.7228558621i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (-0.241 + 0.970i)T \) |
| 3 | \( 1 + (-0.374 + 0.927i)T \) |
| 5 | \( 1 + (0.961 + 0.275i)T \) |
| 7 | \( 1 + (0.0348 + 0.999i)T \) |
| 13 | \( 1 + (0.438 + 0.898i)T \) |
| 17 | \( 1 + (0.438 - 0.898i)T \) |
| 19 | \( 1 + (-0.374 + 0.927i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.978 + 0.207i)T \) |
| 31 | \( 1 + (-0.809 + 0.587i)T \) |
| 41 | \( 1 + (0.990 - 0.139i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.978 - 0.207i)T \) |
| 53 | \( 1 + (0.961 - 0.275i)T \) |
| 59 | \( 1 + (-0.615 + 0.788i)T \) |
| 61 | \( 1 + (-0.719 + 0.694i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (-0.241 - 0.970i)T \) |
| 73 | \( 1 + (0.309 - 0.951i)T \) |
| 79 | \( 1 + (0.559 - 0.829i)T \) |
| 83 | \( 1 + (0.438 - 0.898i)T \) |
| 89 | \( 1 + (-0.939 - 0.342i)T \) |
| 97 | \( 1 + (-0.104 - 0.994i)T \) |
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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
−23.76941226238888905219864618751, −22.84860167376956862798786731401, −22.15382128911695437772607273684, −21.099084269671838202127733668742, −20.274617727349708092989388339988, −19.61879798770898301844969391616, −18.54612879227778120643709023791, −17.812165480981216202739796468657, −17.17962351581207244123781690025, −16.54549986000753325717781926367, −14.56415599562546860745777373234, −13.683011963609068173246784887061, −12.97212958964268740121018325503, −12.54770823339570127578897798705, −11.048231243541138133542996090769, −10.648976790870538873703358105688, −9.561530151698798546121260606210, −8.40453758480567571452288949485, −7.58046106837918292377354862460, −6.276069669071474923305489895191, −5.30770159813239107110733098096, −4.05629218566264449920643453512, −2.66096668077841762088293691674, −1.63667783806934765747019523847, −0.663734270842529475993608142495,
1.75328747437046817746228944859, 3.39972885092405780912429907081, 4.6779093690706307489938175614, 5.75587372357270798845326976690, 5.99565050925022797517206882340, 7.34394557822754872463137907441, 8.84356880075810616412673534385, 9.29557889010824040126083049921, 10.104952229169420181784301252234, 11.173894041549734100249092421617, 12.35201350191235242224852077655, 13.68925562206017089735440383743, 14.48025242574200072442377391642, 15.13014976341904299196723764278, 16.28481791542293036882859880041, 16.583418705028395234082969782128, 17.87236825316929876587968110066, 18.24167736917694889911641917157, 19.29494046635846587504219035410, 20.895279514362794187149981443551, 21.46315827351279699692033000987, 22.300918444503516697560839096975, 22.94320866252319385935704505287, 24.00991436589521454393523041699, 25.01492024000078421906554855145