L(s) = 1 | + (0.926 + 0.377i)2-s + (−0.681 − 0.732i)3-s + (0.715 + 0.698i)4-s + (−0.906 + 0.421i)5-s + (−0.354 − 0.935i)6-s + (0.779 + 0.626i)7-s + (0.399 + 0.916i)8-s + (−0.0724 + 0.997i)9-s + (−0.998 + 0.0483i)10-s + (0.0241 − 0.999i)12-s + (−0.262 − 0.964i)13-s + (0.485 + 0.873i)14-s + (0.926 + 0.377i)15-s + (0.0241 + 0.999i)16-s + (0.568 − 0.822i)17-s + (−0.443 + 0.896i)18-s + ⋯ |
L(s) = 1 | + (0.926 + 0.377i)2-s + (−0.681 − 0.732i)3-s + (0.715 + 0.698i)4-s + (−0.906 + 0.421i)5-s + (−0.354 − 0.935i)6-s + (0.779 + 0.626i)7-s + (0.399 + 0.916i)8-s + (−0.0724 + 0.997i)9-s + (−0.998 + 0.0483i)10-s + (0.0241 − 0.999i)12-s + (−0.262 − 0.964i)13-s + (0.485 + 0.873i)14-s + (0.926 + 0.377i)15-s + (0.0241 + 0.999i)16-s + (0.568 − 0.822i)17-s + (−0.443 + 0.896i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.039429525 + 0.5193793622i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.039429525 + 0.5193793622i\) |
\(L(1)\) |
\(\approx\) |
\(1.426032548 + 0.2301057396i\) |
\(L(1)\) |
\(\approx\) |
\(1.426032548 + 0.2301057396i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (0.926 + 0.377i)T \) |
| 3 | \( 1 + (-0.681 - 0.732i)T \) |
| 5 | \( 1 + (-0.906 + 0.421i)T \) |
| 7 | \( 1 + (0.779 + 0.626i)T \) |
| 13 | \( 1 + (-0.262 - 0.964i)T \) |
| 17 | \( 1 + (0.568 - 0.822i)T \) |
| 19 | \( 1 + (0.0241 - 0.999i)T \) |
| 23 | \( 1 + (0.399 - 0.916i)T \) |
| 29 | \( 1 + (-0.527 + 0.849i)T \) |
| 31 | \( 1 + (0.399 + 0.916i)T \) |
| 37 | \( 1 + (-0.168 - 0.985i)T \) |
| 41 | \( 1 + (0.568 + 0.822i)T \) |
| 43 | \( 1 + (0.485 - 0.873i)T \) |
| 47 | \( 1 + (0.926 + 0.377i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (0.958 + 0.285i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (0.485 + 0.873i)T \) |
| 71 | \( 1 + (0.399 + 0.916i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (-0.906 + 0.421i)T \) |
| 83 | \( 1 + (-0.168 + 0.985i)T \) |
| 89 | \( 1 + (-0.809 - 0.587i)T \) |
| 97 | \( 1 + (0.779 + 0.626i)T \) |
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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
−20.95586627666435843672171832468, −20.24540426585994876245944235781, −19.280008261280194869318515175549, −18.7163832382441538254867024107, −17.22718020846608739482483040429, −16.84887811608359686963724060524, −16.043130813109853890068917181838, −15.24991603861090305009475078563, −14.70169951920078192239440261172, −13.91987329306471958492322637966, −12.864967567240620895288931168444, −12.018148247154895526277554175043, −11.56683446141118295287854778635, −10.95008694101083463488607851400, −10.12538225706486622181431133320, −9.312924971210881019386165586266, −8.02038939537302701075673432326, −7.27783941326463140112506566162, −6.19493780035996793468392982613, −5.38553436095733613561602140184, −4.539661617458968662444327415857, −4.03725548371549820627314481996, −3.436681813990269279162465364879, −1.8290245137216430383393545762, −0.87940264407942825077472359253,
0.91360389199850296356099937748, 2.4207143327140033458441350779, 2.9274365137576205121988457256, 4.24443758581332645423497671439, 5.156671356503138733496944905422, 5.53091665077033530753789947384, 6.792387551626059095559987173, 7.23732677643821974319278611368, 8.00708033550238815027587209849, 8.705660669984516872583408441153, 10.48198464674537748522276798414, 11.206428505217539561519068889410, 11.64375747082710996972567537467, 12.56109143629928807694526885567, 12.82517448841639816898418249927, 14.21441494184309978451875031548, 14.50444570419157912139257620609, 15.605402376872694769208634724206, 15.96364569902392321821152073672, 17.009431604309376901140295566732, 17.77972204271370382736069961249, 18.37789241637425434457485354788, 19.284328935233551407612603207047, 20.091236396321885672409360077890, 20.87501325799833568394788562819