L(s) = 1 | + (0.994 + 0.104i)3-s + (0.978 + 0.207i)9-s + (−0.978 + 0.207i)11-s + (−0.951 + 0.309i)13-s + (−0.406 + 0.913i)17-s + (−0.104 − 0.994i)19-s + (0.743 − 0.669i)23-s + (0.951 + 0.309i)27-s + (−0.809 − 0.587i)29-s + (0.913 + 0.406i)31-s + (−0.994 + 0.104i)33-s + (0.207 − 0.978i)37-s + (−0.978 + 0.207i)39-s + (−0.309 − 0.951i)41-s + i·43-s + ⋯ |
L(s) = 1 | + (0.994 + 0.104i)3-s + (0.978 + 0.207i)9-s + (−0.978 + 0.207i)11-s + (−0.951 + 0.309i)13-s + (−0.406 + 0.913i)17-s + (−0.104 − 0.994i)19-s + (0.743 − 0.669i)23-s + (0.951 + 0.309i)27-s + (−0.809 − 0.587i)29-s + (0.913 + 0.406i)31-s + (−0.994 + 0.104i)33-s + (0.207 − 0.978i)37-s + (−0.978 + 0.207i)39-s + (−0.309 − 0.951i)41-s + i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.804439441 - 0.2339608980i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.804439441 - 0.2339608980i\) |
\(L(1)\) |
\(\approx\) |
\(1.411505217 + 0.02910691424i\) |
\(L(1)\) |
\(\approx\) |
\(1.411505217 + 0.02910691424i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.994 + 0.104i)T \) |
| 11 | \( 1 + (-0.978 + 0.207i)T \) |
| 13 | \( 1 + (-0.951 + 0.309i)T \) |
| 17 | \( 1 + (-0.406 + 0.913i)T \) |
| 19 | \( 1 + (-0.104 - 0.994i)T \) |
| 23 | \( 1 + (0.743 - 0.669i)T \) |
| 29 | \( 1 + (-0.809 - 0.587i)T \) |
| 31 | \( 1 + (0.913 + 0.406i)T \) |
| 37 | \( 1 + (0.207 - 0.978i)T \) |
| 41 | \( 1 + (-0.309 - 0.951i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (-0.406 - 0.913i)T \) |
| 53 | \( 1 + (0.994 + 0.104i)T \) |
| 59 | \( 1 + (0.669 - 0.743i)T \) |
| 61 | \( 1 + (0.669 + 0.743i)T \) |
| 67 | \( 1 + (0.406 - 0.913i)T \) |
| 71 | \( 1 + (0.809 + 0.587i)T \) |
| 73 | \( 1 + (0.207 + 0.978i)T \) |
| 79 | \( 1 + (0.913 - 0.406i)T \) |
| 83 | \( 1 + (0.587 + 0.809i)T \) |
| 89 | \( 1 + (0.669 + 0.743i)T \) |
| 97 | \( 1 + (0.587 - 0.809i)T \) |
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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
−20.611944336840099832852791090900, −19.95826477713000887196879034563, −19.06826013708508247737941730592, −18.5709285381266200500795598016, −17.77329410859055350058402766812, −16.77682090457988825318888049402, −15.93959448611769771846604907456, −15.173750352899025393629378847, −14.644362480474134224538546341917, −13.64963757530306361678180717294, −13.1789629398865093271712777220, −12.348315059573733429949099369323, −11.4294374407984370984730760349, −10.32195852278693730401977462837, −9.76420118290420264074048343336, −8.92902412528496985115721604771, −7.997112353200513043531189696875, −7.51078240443102970235536874303, −6.62559697857565693101966269579, −5.36356320365212586567186681572, −4.66679202391027907769046512338, −3.49437178277001920232845593748, −2.77676320148571934076012340862, −1.99098764166410230634069204074, −0.744917596486339099771828919,
0.60682173423604797276391531671, 2.14849591821887072618240285798, 2.48825626636947115429339000958, 3.64019193797445315939410131879, 4.5506075476503172633367523904, 5.25313404475067851582193244146, 6.638099733676297178650663470765, 7.29999141753797209615748015176, 8.15445178395271424460712298223, 8.84201212661186474940906857533, 9.689542609526768345831935027026, 10.39062080888295459840733316130, 11.20453434683123734269148265594, 12.431301073887938167285309726205, 13.01492165290157873105988013552, 13.674428847414240368911905449539, 14.663575237237090082884725286963, 15.15581351172768179601890174774, 15.819806094318041143043157833465, 16.796932938604703047927563497348, 17.62354220396379428180210290316, 18.45863926522219688718825134524, 19.31657333265775334765055124266, 19.69136313456868020940584738952, 20.62860452738975248769526290596