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
−20.71730695946704899290554042735, −19.75046595952386278618015702896, −19.02507902916521787311305675935, −18.39032909477127596801924014158, −17.55547145103615954287938595139, −17.16121198120326085751054494534, −16.07183376342304305979272427722, −15.93869562467866993409902645342, −15.11164898702995354681238780556, −13.87036493866225988649730705484, −12.85078812244694594417886946118, −12.08830734331636688992305399040, −11.38013166163911435418234383069, −10.978104078022115020048710282959, −9.55156052736396147850955221801, −9.17460283887351264146390214879, −8.45262793662783838299391217485, −7.31318768693461715683680483239, −6.731733964434861654363221350544, −5.69177681799784130435435823186, −5.16708706904305068366235338914, −4.307797731402416233396623269457, −2.525066357644912916731370204501, −1.633695856447018226112002354702, −0.65605221444624839280589938193,
0.65413598117217902262120909482, 1.645750908680028649868724810225, 2.97906387629928451234785179188, 3.733056063939663008569149600358, 4.72917692392930299063812397993, 6.14404592237379649259074677382, 6.59377025132584515835536512397, 7.50905282441128327962048622990, 8.037696520730191622279179750780, 9.363504334514231559058116893528, 10.338253292751588362720432981168, 10.59461377468762476104012939983, 11.15206595817289377414632558802, 12.05990215223939512539660159237, 12.82250415131153142527119096028, 13.79028444319104537299799304821, 14.99892184537183598009740981961, 15.60028053556104276008717102267, 16.397712776775659290624567048555, 17.23991984099454402249615570985, 17.669746538425051433941190714351, 18.325759941198712986834659755981, 19.09044405703179912038374774751, 19.85279625916898032122366990376, 20.669653320053415217034020871023