L(s) = 1 | + (0.669 − 0.743i)2-s + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (0.669 + 0.743i)5-s + (0.309 − 0.951i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + 10-s + (−0.5 − 0.866i)12-s + (−0.978 − 0.207i)13-s + (−0.809 − 0.587i)14-s + (0.913 + 0.406i)15-s + (−0.978 + 0.207i)16-s + (−0.978 + 0.207i)17-s + (−0.104 − 0.994i)18-s + ⋯ |
L(s) = 1 | + (0.669 − 0.743i)2-s + (0.913 − 0.406i)3-s + (−0.104 − 0.994i)4-s + (0.669 + 0.743i)5-s + (0.309 − 0.951i)6-s + (−0.104 − 0.994i)7-s + (−0.809 − 0.587i)8-s + (0.669 − 0.743i)9-s + 10-s + (−0.5 − 0.866i)12-s + (−0.978 − 0.207i)13-s + (−0.809 − 0.587i)14-s + (0.913 + 0.406i)15-s + (−0.978 + 0.207i)16-s + (−0.978 + 0.207i)17-s + (−0.104 − 0.994i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.487506653 - 2.132668442i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.487506653 - 2.132668442i\) |
\(L(1)\) |
\(\approx\) |
\(1.588748598 - 1.181321911i\) |
\(L(1)\) |
\(\approx\) |
\(1.588748598 - 1.181321911i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (0.669 - 0.743i)T \) |
| 3 | \( 1 + (0.913 - 0.406i)T \) |
| 5 | \( 1 + (0.669 + 0.743i)T \) |
| 7 | \( 1 + (-0.104 - 0.994i)T \) |
| 13 | \( 1 + (-0.978 - 0.207i)T \) |
| 17 | \( 1 + (-0.978 + 0.207i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.309 + 0.951i)T \) |
| 41 | \( 1 + (0.913 - 0.406i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (-0.809 - 0.587i)T \) |
| 53 | \( 1 + (0.669 - 0.743i)T \) |
| 59 | \( 1 + (0.913 + 0.406i)T \) |
| 61 | \( 1 + (0.669 + 0.743i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (0.669 + 0.743i)T \) |
| 73 | \( 1 + (-0.809 + 0.587i)T \) |
| 79 | \( 1 + (-0.978 - 0.207i)T \) |
| 83 | \( 1 + (-0.978 + 0.207i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.309 + 0.951i)T \) |
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
−24.53671276711869347022534643565, −24.33589516221509926849335340006, −22.56025570546063988719804075297, −22.05113964779140663033279840938, −21.15445197892475801708942793305, −20.64831889540570203002744481029, −19.53992097751416518047066516558, −18.39080735784456589474615687251, −17.36182864963774049532339036682, −16.45266377999037353401261235115, −15.654281825673976785282857783651, −14.90587198731501644846884380624, −14.08667520778676888870075321564, −13.186354134046228320881527060806, −12.567849650835487493439198547865, −11.42416148853879318930542207167, −9.58912376267602972442272857995, −9.22262762304309187396239684970, −8.25823673503896176099527379151, −7.28683145951480578408008268242, −5.97054221814794761745598527416, −5.068637153935997359806173567019, −4.30687069340977037979254087331, −2.87857973594147747935427630852, −2.12314352620896458717734849770,
1.18882718426305384039298000504, 2.38004309832875287143608498546, 3.108032845623124812638005553437, 4.144209339980594592537874925711, 5.381123524064279413836679936432, 6.84936189814790519171980799705, 7.20158113526815381728996784679, 8.939707837967848595463569952961, 9.80663847188385493799346488600, 10.53373546100836225341023083384, 11.53103424687902650353532241269, 12.93924620865475833367467383175, 13.28988836267256412468595257102, 14.31697884290858219426924564482, 14.66872906666982758643980184536, 15.79147636185529455833184769861, 17.44125392145551410402597592576, 18.10810485374463685338562418762, 19.22847943195061318340497908750, 19.74737303215641330605773012602, 20.56062802408411774006256096919, 21.38335564448893412082460781994, 22.29110495659393039425536842938, 22.99899431416203290128858357491, 24.16925235949475842106594444567