L(s) = 1 | + (−0.0307 − 0.999i)2-s + (−0.998 + 0.0615i)4-s + (0.779 + 0.626i)5-s + (0.816 + 0.577i)7-s + (0.0922 + 0.995i)8-s + (0.602 − 0.798i)10-s + (0.0307 + 0.999i)11-s + (−0.908 + 0.417i)13-s + (0.552 − 0.833i)14-s + (0.992 − 0.122i)16-s + (0.739 − 0.673i)17-s + (−0.273 + 0.961i)19-s + (−0.816 − 0.577i)20-s + (0.998 − 0.0615i)22-s + (−0.0307 + 0.999i)23-s + ⋯ |
L(s) = 1 | + (−0.0307 − 0.999i)2-s + (−0.998 + 0.0615i)4-s + (0.779 + 0.626i)5-s + (0.816 + 0.577i)7-s + (0.0922 + 0.995i)8-s + (0.602 − 0.798i)10-s + (0.0307 + 0.999i)11-s + (−0.908 + 0.417i)13-s + (0.552 − 0.833i)14-s + (0.992 − 0.122i)16-s + (0.739 − 0.673i)17-s + (−0.273 + 0.961i)19-s + (−0.816 − 0.577i)20-s + (0.998 − 0.0615i)22-s + (−0.0307 + 0.999i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.487823301 + 1.184604778i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.487823301 + 1.184604778i\) |
\(L(1)\) |
\(\approx\) |
\(1.138377727 - 0.04722602131i\) |
\(L(1)\) |
\(\approx\) |
\(1.138377727 - 0.04722602131i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 103 | \( 1 \) |
good | 2 | \( 1 + (-0.0307 - 0.999i)T \) |
| 5 | \( 1 + (0.779 + 0.626i)T \) |
| 7 | \( 1 + (0.816 + 0.577i)T \) |
| 11 | \( 1 + (0.0307 + 0.999i)T \) |
| 13 | \( 1 + (-0.908 + 0.417i)T \) |
| 17 | \( 1 + (0.739 - 0.673i)T \) |
| 19 | \( 1 + (-0.273 + 0.961i)T \) |
| 23 | \( 1 + (-0.0307 + 0.999i)T \) |
| 29 | \( 1 + (-0.153 + 0.988i)T \) |
| 31 | \( 1 + (0.389 - 0.920i)T \) |
| 37 | \( 1 + (0.982 + 0.183i)T \) |
| 41 | \( 1 + (-0.153 - 0.988i)T \) |
| 43 | \( 1 + (-0.332 + 0.943i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.273 - 0.961i)T \) |
| 59 | \( 1 + (0.816 - 0.577i)T \) |
| 61 | \( 1 + (0.213 + 0.976i)T \) |
| 67 | \( 1 + (-0.816 + 0.577i)T \) |
| 71 | \( 1 + (-0.932 - 0.361i)T \) |
| 73 | \( 1 + (-0.932 - 0.361i)T \) |
| 79 | \( 1 + (-0.153 + 0.988i)T \) |
| 83 | \( 1 + (0.816 + 0.577i)T \) |
| 89 | \( 1 + (-0.445 - 0.895i)T \) |
| 97 | \( 1 + (-0.952 + 0.303i)T \) |
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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
−21.64998278824460961014580517046, −20.89693987895865309278241409435, −19.86995786163823260438218208542, −18.98900236709600379036825135259, −18.00999741551571433925649312010, −17.28622048764289161398770766372, −16.89477243748843049488929043304, −16.13440470555440523963921743507, −14.98710181207382071059657202346, −14.38187005822813906893809725827, −13.62529128765934514481501839099, −12.98292569470720218964304880152, −12.01767090554064836927146859436, −10.67531133418062771330284599554, −10.02064903684063817792658156312, −8.985393059098606706134759775630, −8.299619994166809840124494192546, −7.58845577111400515425896982343, −6.46799112312891878687702713597, −5.69043344915611972750125290734, −4.87618781645009921221173898584, −4.18864573016889098891908719481, −2.72908906988276351923657049001, −1.24176034089432089634726206907, −0.42855727776466387486841381014,
1.39603588311022184256857536373, 2.089908183185779803312539485464, 2.83581239957681760143135015925, 4.0644896186058831972832354099, 5.09353143984261821942728040832, 5.685392990966799580336584021833, 7.12649100920014795069268924807, 7.931774236515671048910431743199, 9.13095366384004236844393221594, 9.78711737956603861607261937039, 10.35254643408109917438226217331, 11.50349035497012185814777276658, 11.963618150325438190465719938768, 12.87469477792548343594804259874, 13.824999771022695077311899765791, 14.68805637430320769671713871548, 14.890464630867252298979094275628, 16.58999932416463486434808120367, 17.4540524226587808387608721196, 18.01033984065494842749982827022, 18.66596103382928157798381230578, 19.39705372749540819205634002402, 20.51810673491157779583049747187, 20.993745692443749083137335645294, 21.768581997119485584458323118290