L(s) = 1 | + (−0.943 + 0.331i)2-s + (−0.989 − 0.144i)3-s + (0.779 − 0.626i)4-s + (−0.168 − 0.985i)5-s + (0.981 − 0.192i)6-s + (0.120 + 0.992i)7-s + (−0.527 + 0.849i)8-s + (0.958 + 0.285i)9-s + (0.485 + 0.873i)10-s + (−0.861 + 0.506i)12-s + (0.120 + 0.992i)13-s + (−0.443 − 0.896i)14-s + (0.0241 + 0.999i)15-s + (0.215 − 0.976i)16-s + (−0.861 − 0.506i)17-s + (−0.998 + 0.0483i)18-s + ⋯ |
L(s) = 1 | + (−0.943 + 0.331i)2-s + (−0.989 − 0.144i)3-s + (0.779 − 0.626i)4-s + (−0.168 − 0.985i)5-s + (0.981 − 0.192i)6-s + (0.120 + 0.992i)7-s + (−0.527 + 0.849i)8-s + (0.958 + 0.285i)9-s + (0.485 + 0.873i)10-s + (−0.861 + 0.506i)12-s + (0.120 + 0.992i)13-s + (−0.443 − 0.896i)14-s + (0.0241 + 0.999i)15-s + (0.215 − 0.976i)16-s + (−0.861 − 0.506i)17-s + (−0.998 + 0.0483i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6217546375 + 0.01226750510i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6217546375 + 0.01226750510i\) |
\(L(1)\) |
\(\approx\) |
\(0.5354410893 + 0.01523179976i\) |
\(L(1)\) |
\(\approx\) |
\(0.5354410893 + 0.01523179976i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (-0.943 + 0.331i)T \) |
| 3 | \( 1 + (-0.989 - 0.144i)T \) |
| 5 | \( 1 + (-0.168 - 0.985i)T \) |
| 7 | \( 1 + (0.120 + 0.992i)T \) |
| 13 | \( 1 + (0.120 + 0.992i)T \) |
| 17 | \( 1 + (-0.861 - 0.506i)T \) |
| 19 | \( 1 + (0.399 - 0.916i)T \) |
| 23 | \( 1 + (0.926 + 0.377i)T \) |
| 29 | \( 1 + (0.568 + 0.822i)T \) |
| 31 | \( 1 + (-0.0724 - 0.997i)T \) |
| 37 | \( 1 + (-0.354 - 0.935i)T \) |
| 41 | \( 1 + (0.995 + 0.0965i)T \) |
| 43 | \( 1 + (-0.168 - 0.985i)T \) |
| 47 | \( 1 + (0.958 + 0.285i)T \) |
| 53 | \( 1 + (-0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.861 - 0.506i)T \) |
| 61 | \( 1 + (0.309 + 0.951i)T \) |
| 67 | \( 1 + (-0.168 + 0.985i)T \) |
| 71 | \( 1 + (0.644 + 0.764i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (-0.443 + 0.896i)T \) |
| 83 | \( 1 + (0.779 + 0.626i)T \) |
| 89 | \( 1 + (0.309 - 0.951i)T \) |
| 97 | \( 1 + (-0.906 + 0.421i)T \) |
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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
−20.669653320053415217034020871023, −19.85279625916898032122366990376, −19.09044405703179912038374774751, −18.325759941198712986834659755981, −17.669746538425051433941190714351, −17.23991984099454402249615570985, −16.397712776775659290624567048555, −15.60028053556104276008717102267, −14.99892184537183598009740981961, −13.79028444319104537299799304821, −12.82250415131153142527119096028, −12.05990215223939512539660159237, −11.15206595817289377414632558802, −10.59461377468762476104012939983, −10.338253292751588362720432981168, −9.363504334514231559058116893528, −8.037696520730191622279179750780, −7.50905282441128327962048622990, −6.59377025132584515835536512397, −6.14404592237379649259074677382, −4.72917692392930299063812397993, −3.733056063939663008569149600358, −2.97906387629928451234785179188, −1.645750908680028649868724810225, −0.65413598117217902262120909482,
0.65605221444624839280589938193, 1.633695856447018226112002354702, 2.525066357644912916731370204501, 4.307797731402416233396623269457, 5.16708706904305068366235338914, 5.69177681799784130435435823186, 6.731733964434861654363221350544, 7.31318768693461715683680483239, 8.45262793662783838299391217485, 9.17460283887351264146390214879, 9.55156052736396147850955221801, 10.978104078022115020048710282959, 11.38013166163911435418234383069, 12.08830734331636688992305399040, 12.85078812244694594417886946118, 13.87036493866225988649730705484, 15.11164898702995354681238780556, 15.93869562467866993409902645342, 16.07183376342304305979272427722, 17.16121198120326085751054494534, 17.55547145103615954287938595139, 18.39032909477127596801924014158, 19.02507902916521787311305675935, 19.75046595952386278618015702896, 20.71730695946704899290554042735