L(s) = 1 | + (0.203 − 0.979i)2-s + (0.854 − 0.519i)3-s + (−0.917 − 0.398i)4-s + (0.682 + 0.730i)5-s + (−0.334 − 0.942i)6-s + (−0.0682 − 0.997i)7-s + (−0.576 + 0.816i)8-s + (0.460 − 0.887i)9-s + (0.854 − 0.519i)10-s + (0.460 − 0.887i)11-s + (−0.990 + 0.136i)12-s + (0.460 + 0.887i)13-s + (−0.990 − 0.136i)14-s + (0.962 + 0.269i)15-s + (0.682 + 0.730i)16-s + (0.962 − 0.269i)17-s + ⋯ |
L(s) = 1 | + (0.203 − 0.979i)2-s + (0.854 − 0.519i)3-s + (−0.917 − 0.398i)4-s + (0.682 + 0.730i)5-s + (−0.334 − 0.942i)6-s + (−0.0682 − 0.997i)7-s + (−0.576 + 0.816i)8-s + (0.460 − 0.887i)9-s + (0.854 − 0.519i)10-s + (0.460 − 0.887i)11-s + (−0.990 + 0.136i)12-s + (0.460 + 0.887i)13-s + (−0.990 − 0.136i)14-s + (0.962 + 0.269i)15-s + (0.682 + 0.730i)16-s + (0.962 − 0.269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.177718167 - 2.185839138i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.177718167 - 2.185839138i\) |
\(L(1)\) |
\(\approx\) |
\(1.256437467 - 1.104440497i\) |
\(L(1)\) |
\(\approx\) |
\(1.256437467 - 1.104440497i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (0.203 - 0.979i)T \) |
| 3 | \( 1 + (0.854 - 0.519i)T \) |
| 5 | \( 1 + (0.682 + 0.730i)T \) |
| 7 | \( 1 + (-0.0682 - 0.997i)T \) |
| 11 | \( 1 + (0.460 - 0.887i)T \) |
| 13 | \( 1 + (0.460 + 0.887i)T \) |
| 17 | \( 1 + (0.962 - 0.269i)T \) |
| 19 | \( 1 + (0.854 - 0.519i)T \) |
| 23 | \( 1 + (-0.917 - 0.398i)T \) |
| 29 | \( 1 + (0.460 + 0.887i)T \) |
| 31 | \( 1 + (-0.576 + 0.816i)T \) |
| 37 | \( 1 + (0.460 + 0.887i)T \) |
| 41 | \( 1 + (-0.334 - 0.942i)T \) |
| 43 | \( 1 + (0.962 + 0.269i)T \) |
| 47 | \( 1 + (0.962 + 0.269i)T \) |
| 53 | \( 1 + (-0.917 - 0.398i)T \) |
| 59 | \( 1 + (-0.917 + 0.398i)T \) |
| 61 | \( 1 + (-0.334 - 0.942i)T \) |
| 67 | \( 1 + (-0.576 - 0.816i)T \) |
| 71 | \( 1 + (0.203 + 0.979i)T \) |
| 73 | \( 1 + (-0.917 + 0.398i)T \) |
| 79 | \( 1 + (-0.917 + 0.398i)T \) |
| 83 | \( 1 + (-0.334 - 0.942i)T \) |
| 89 | \( 1 + (-0.576 - 0.816i)T \) |
| 97 | \( 1 + T \) |
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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
−22.074489191105585344507256383242, −21.268968358597781179181234180, −20.641979643850024768980420878606, −19.79460585666134375248953242329, −18.66405579309462050692558535067, −17.99432491141171533785210813236, −17.160083653214278521268687689701, −16.24413514043449323283855298921, −15.68403872169814658802442778148, −14.91759386926162833045424768526, −14.25187625913806226389774025789, −13.45443441599293235810676252496, −12.65137774074515739821955197076, −12.01212545298034946630726379504, −10.18782076018390500866319385295, −9.56509608193236337788800822018, −9.04856201171947485411765994347, −8.06690322866321855992111837567, −7.60302396272901817225970269373, −5.96313101627815305192173879295, −5.63494389268344493636320007870, −4.60086532606147656419319960662, −3.74212363779750652069639328514, −2.64792312128189389897439322005, −1.44160652798519862561527952507,
1.052495416291011887390491787112, 1.71253077764963173394301661695, 3.001464503616713055897515587168, 3.393432707338409492282607305348, 4.39223883365787058806171743926, 5.79663567819101507475661596225, 6.66853363537566781634815034442, 7.5574510910368536101393808726, 8.70077977555767267513516452537, 9.41728487849262516811517413120, 10.17595110451676811139314174240, 10.98819117839827505205264576926, 11.83785023930670097496806970624, 12.811044164737954708501881004266, 13.784160514252815182322290809444, 14.09635183920763130211945418481, 14.3840933555952883171316885352, 15.887187890138441752958146323817, 16.99489732519861699866659298685, 17.920330993300505665254842487519, 18.61599541484128472679048711694, 19.10922451389962920351866938010, 20.01387493092045093985760504334, 20.55635994778349269405464074902, 21.45050053051637772462839730752