L(s) = 1 | + (−0.993 − 0.116i)2-s + (−0.951 − 0.307i)3-s + (0.972 + 0.232i)4-s + (0.763 − 0.646i)5-s + (0.909 + 0.416i)6-s + (0.241 − 0.970i)7-s + (−0.938 − 0.344i)8-s + (0.811 + 0.584i)9-s + (−0.833 + 0.552i)10-s + (−0.996 + 0.0779i)11-s + (−0.854 − 0.519i)12-s + (−0.511 + 0.859i)13-s + (−0.353 + 0.935i)14-s + (−0.924 + 0.380i)15-s + (0.892 + 0.451i)16-s + (0.917 + 0.398i)17-s + ⋯ |
L(s) = 1 | + (−0.993 − 0.116i)2-s + (−0.951 − 0.307i)3-s + (0.972 + 0.232i)4-s + (0.763 − 0.646i)5-s + (0.909 + 0.416i)6-s + (0.241 − 0.970i)7-s + (−0.938 − 0.344i)8-s + (0.811 + 0.584i)9-s + (−0.833 + 0.552i)10-s + (−0.996 + 0.0779i)11-s + (−0.854 − 0.519i)12-s + (−0.511 + 0.859i)13-s + (−0.353 + 0.935i)14-s + (−0.924 + 0.380i)15-s + (0.892 + 0.451i)16-s + (0.917 + 0.398i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1016449056 - 0.4310184799i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1016449056 - 0.4310184799i\) |
\(L(1)\) |
\(\approx\) |
\(0.4968074589 - 0.1754637899i\) |
\(L(1)\) |
\(\approx\) |
\(0.4968074589 - 0.1754637899i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 \) |
| 139 | \( 1 \) |
good | 2 | \( 1 + (-0.993 - 0.116i)T \) |
| 3 | \( 1 + (-0.951 - 0.307i)T \) |
| 5 | \( 1 + (0.763 - 0.646i)T \) |
| 7 | \( 1 + (0.241 - 0.970i)T \) |
| 11 | \( 1 + (-0.996 + 0.0779i)T \) |
| 13 | \( 1 + (-0.511 + 0.859i)T \) |
| 17 | \( 1 + (0.917 + 0.398i)T \) |
| 19 | \( 1 + (-0.999 + 0.0390i)T \) |
| 23 | \( 1 + (-0.999 - 0.0195i)T \) |
| 31 | \( 1 + (-0.0487 + 0.998i)T \) |
| 37 | \( 1 + (0.998 - 0.0585i)T \) |
| 41 | \( 1 + (0.0682 + 0.997i)T \) |
| 43 | \( 1 + (0.222 - 0.974i)T \) |
| 47 | \( 1 + (0.407 + 0.913i)T \) |
| 53 | \( 1 + (0.993 + 0.116i)T \) |
| 59 | \( 1 + (-0.775 + 0.631i)T \) |
| 61 | \( 1 + (-0.494 + 0.869i)T \) |
| 67 | \( 1 + (0.996 + 0.0779i)T \) |
| 71 | \( 1 + (0.981 + 0.193i)T \) |
| 73 | \( 1 + (-0.316 - 0.948i)T \) |
| 79 | \( 1 + (0.822 - 0.568i)T \) |
| 83 | \( 1 + (-0.932 - 0.362i)T \) |
| 89 | \( 1 + (0.668 - 0.744i)T \) |
| 97 | \( 1 + (0.900 - 0.433i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.448929459910436281127615504235, −18.19574374418569421930365689554, −17.427197741765198065489976092837, −16.95608597823977106885484714953, −16.14716642678123612536968655076, −15.408756703044451375462378992227, −15.04948128131376073817910213589, −14.27652190870639951437693100449, −13.031563635408669472122977636403, −12.4110719016250964498423589297, −11.719277523484354127008248148324, −10.91498636167249115734775069878, −10.49178114377178996126372075970, −9.75582841071071397056191682952, −9.41558621483150567046128276673, −8.19116256059409874692640003988, −7.71328738152286818929996809466, −6.773406605323776027238764121423, −5.979573712574118619465155925248, −5.61636084370404522331785453891, −4.99629546341050527975183079976, −3.557867145889532883117210276939, −2.464227559877947435115606507873, −2.19185915155278488544328847515, −0.88696019730024324488642344218,
0.24109674716948790210738789746, 1.23640111458338308630661175198, 1.80337284105854415394309032721, 2.61335796410570680561428219861, 4.047838393511279061811432188544, 4.737322515433098376188893707579, 5.66986386190250638084610637599, 6.25312371198941616438051277106, 7.04507016743508060542800600915, 7.733104544326029477831636848106, 8.30422985694887493544725050292, 9.326861960715577676039111308744, 10.125549315362812299796120272122, 10.399584388934073581898954622195, 11.078164570133250846044627982634, 12.04623386686800729044027578208, 12.48313912656848502171543279267, 13.20436457875437559690489473942, 13.98280579977983385016756929210, 14.8646768022903022916957392434, 15.969502864248508685877958367385, 16.48667097242672716907195055505, 16.91110087087882768953179496388, 17.44327214661031855863707917756, 18.073754798036250519083279188840