L(s) = 1 | + (0.923 − 0.382i)3-s + (−0.382 + 0.923i)5-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)9-s + (−0.923 − 0.382i)11-s + (−0.382 − 0.923i)13-s + i·15-s − i·17-s + (0.382 + 0.923i)19-s + (0.923 + 0.382i)21-s + (−0.707 + 0.707i)23-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s + (−0.923 + 0.382i)29-s − 31-s + ⋯ |
L(s) = 1 | + (0.923 − 0.382i)3-s + (−0.382 + 0.923i)5-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)9-s + (−0.923 − 0.382i)11-s + (−0.382 − 0.923i)13-s + i·15-s − i·17-s + (0.382 + 0.923i)19-s + (0.923 + 0.382i)21-s + (−0.707 + 0.707i)23-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s + (−0.923 + 0.382i)29-s − 31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.139296592 + 0.05597005255i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.139296592 + 0.05597005255i\) |
\(L(1)\) |
\(\approx\) |
\(1.232080730 + 0.02781530869i\) |
\(L(1)\) |
\(\approx\) |
\(1.232080730 + 0.02781530869i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (0.923 - 0.382i)T \) |
| 5 | \( 1 + (-0.382 + 0.923i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
| 11 | \( 1 + (-0.923 - 0.382i)T \) |
| 13 | \( 1 + (-0.382 - 0.923i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (0.382 + 0.923i)T \) |
| 23 | \( 1 + (-0.707 + 0.707i)T \) |
| 29 | \( 1 + (-0.923 + 0.382i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.382 - 0.923i)T \) |
| 41 | \( 1 + (-0.707 + 0.707i)T \) |
| 43 | \( 1 + (0.923 + 0.382i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.923 - 0.382i)T \) |
| 59 | \( 1 + (-0.382 + 0.923i)T \) |
| 61 | \( 1 + (0.923 - 0.382i)T \) |
| 67 | \( 1 + (0.923 - 0.382i)T \) |
| 71 | \( 1 + (0.707 + 0.707i)T \) |
| 73 | \( 1 + (0.707 - 0.707i)T \) |
| 79 | \( 1 + iT \) |
| 83 | \( 1 + (0.382 + 0.923i)T \) |
| 89 | \( 1 + (-0.707 - 0.707i)T \) |
| 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
−32.23277118107529599834800281073, −31.15282655854938056820826254279, −30.46905657956390289506583176310, −28.775081442741461935223219464044, −27.745922434936229049540609237191, −26.6386513574994404172130068103, −25.834189527740095073044408411548, −24.20264355956364454234596804088, −23.84618068584189719160611871492, −21.867930753557755275890635505234, −20.70380013162467957166132166112, −20.16220939388545075440437868505, −18.9134449898731447334631274588, −17.26104379380408616490157998275, −16.11369456761171188567158556742, −14.98264605708834905268156112077, −13.772752614546248389703923276805, −12.66116385235673894014680508664, −10.991264612932908055313704842530, −9.60325088205408122465296497427, −8.352469010463439010624051097960, −7.4253340202776252554441693850, −4.9260467942592083105867321299, −4.0048916562852408839103641437, −1.94793471944494086816269791353,
2.27752717834523380918554012100, 3.42968428235922852394869272527, 5.53845643692668285308686015481, 7.426092800465005318973478490809, 8.11771473278443569354260868236, 9.732696330216818831541813382087, 11.203363691771786506561929443241, 12.549208560747391273181108350394, 13.99123185575273673402839645659, 14.90328890730817422428685867451, 15.837667224396152007788729301210, 18.17069000106866864028836396650, 18.43488168960297054128202120169, 19.840703973405342560743660432843, 20.93647603389952435776181753355, 22.14267124240086251764129525029, 23.52043346483644352524245125627, 24.65338290380527727790769103504, 25.605745502505357816358877647958, 26.75727318256883526562718742384, 27.54094863023883067142901864575, 29.35107762511511312731455676498, 30.19701022949553849381931632575, 31.380345561853704940477793858191, 31.72767436844139208738423911267