L(s) = 1 | + (−0.995 + 0.0920i)2-s + (−0.926 + 0.375i)3-s + (0.983 − 0.183i)4-s + (0.630 − 0.775i)5-s + (0.888 − 0.459i)6-s + (−0.899 − 0.437i)7-s + (−0.962 + 0.272i)8-s + (0.717 − 0.696i)9-s + (−0.556 + 0.830i)10-s + (−0.604 − 0.796i)11-s + (−0.842 + 0.539i)12-s + (0.549 − 0.835i)13-s + (0.935 + 0.352i)14-s + (−0.293 + 0.956i)15-s + (0.932 − 0.360i)16-s + (−0.433 + 0.901i)17-s + ⋯ |
L(s) = 1 | + (−0.995 + 0.0920i)2-s + (−0.926 + 0.375i)3-s + (0.983 − 0.183i)4-s + (0.630 − 0.775i)5-s + (0.888 − 0.459i)6-s + (−0.899 − 0.437i)7-s + (−0.962 + 0.272i)8-s + (0.717 − 0.696i)9-s + (−0.556 + 0.830i)10-s + (−0.604 − 0.796i)11-s + (−0.842 + 0.539i)12-s + (0.549 − 0.835i)13-s + (0.935 + 0.352i)14-s + (−0.293 + 0.956i)15-s + (0.932 − 0.360i)16-s + (−0.433 + 0.901i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 751 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6778313664 - 0.1678080652i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6778313664 - 0.1678080652i\) |
\(L(1)\) |
\(\approx\) |
\(0.5150167668 - 0.05169804306i\) |
\(L(1)\) |
\(\approx\) |
\(0.5150167668 - 0.05169804306i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 751 | \( 1 \) |
good | 2 | \( 1 + (-0.995 + 0.0920i)T \) |
| 3 | \( 1 + (-0.926 + 0.375i)T \) |
| 5 | \( 1 + (0.630 - 0.775i)T \) |
| 7 | \( 1 + (-0.899 - 0.437i)T \) |
| 11 | \( 1 + (-0.604 - 0.796i)T \) |
| 13 | \( 1 + (0.549 - 0.835i)T \) |
| 17 | \( 1 + (-0.433 + 0.901i)T \) |
| 19 | \( 1 + (-0.923 - 0.383i)T \) |
| 23 | \( 1 + (-0.687 + 0.726i)T \) |
| 29 | \( 1 + (0.767 + 0.640i)T \) |
| 31 | \( 1 + (0.440 + 0.897i)T \) |
| 37 | \( 1 + (-0.410 + 0.911i)T \) |
| 41 | \( 1 + (0.992 - 0.125i)T \) |
| 43 | \( 1 + (0.998 + 0.0502i)T \) |
| 47 | \( 1 + (0.932 + 0.360i)T \) |
| 53 | \( 1 + (-0.425 + 0.904i)T \) |
| 59 | \( 1 + (0.813 + 0.580i)T \) |
| 61 | \( 1 + (-0.570 - 0.821i)T \) |
| 67 | \( 1 + (0.846 - 0.532i)T \) |
| 71 | \( 1 + (0.656 + 0.754i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.923 - 0.383i)T \) |
| 83 | \( 1 + (-0.832 - 0.553i)T \) |
| 89 | \( 1 + (-0.455 + 0.890i)T \) |
| 97 | \( 1 + (-0.0711 + 0.997i)T \) |
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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
−22.441404572644708037085923512623, −21.38756141119671635418981795559, −20.82333697409578985321795504817, −19.44080966220931998496340142477, −18.85826848167156769904057917795, −18.21747203803459260996504756526, −17.691171507830329824158290378207, −16.775629741440545303548993915949, −15.984473944258247992730256980548, −15.378956060683054086089510078883, −14.064149824889015761817856049110, −12.97660344105526403924391156620, −12.25523090067547834249226299968, −11.36380996758587685706342040694, −10.54418244456053376428324593515, −9.917488506362988899326333167, −9.14018188822114318917199964410, −7.84905070819527517428657098757, −6.85156272659609378910295917360, −6.41648117630514239529347993760, −5.65555584972738314687851558810, −4.11499903644032581164457829421, −2.42685699889363327058141823009, −2.14350489340447170182798568501, −0.522826405752120138226560844596,
0.51422322977251849136120456781, 1.293616036868909572393679225913, 2.805562097081147289622115609599, 4.02890343499932238446773169458, 5.40172154689138346097442980005, 6.0657231221360004801478068631, 6.69004664830565377628169687778, 8.06185915360215747234596440129, 8.85899094208332799325069504952, 9.74820499363034687203205612331, 10.59005317020265469167221736006, 10.86352926054970999560461839719, 12.3286104610553082997473271834, 12.826522358703291314305901169078, 13.86094635113685389680982634815, 15.53297672338091668427443592424, 15.82760508357074939670669677269, 16.61611929372712473046424052743, 17.39129712800150691738231478588, 17.77475113957318576505416820029, 18.85912396677352490850060261562, 19.71469884141575305790646555491, 20.51805709656023323745039229538, 21.40214704506937185656945337126, 21.88821518622403196548326395302