L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.866 + 0.5i)5-s + 7-s + (0.5 − 0.866i)9-s + i·11-s + (0.866 + 0.5i)13-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)21-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s − i·27-s + (0.866 + 0.5i)29-s + 31-s + (0.5 + 0.866i)33-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (−0.866 + 0.5i)5-s + 7-s + (0.5 − 0.866i)9-s + i·11-s + (0.866 + 0.5i)13-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.866 − 0.5i)21-s + (−0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s − i·27-s + (0.866 + 0.5i)29-s + 31-s + (0.5 + 0.866i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.658117478 + 0.02736754357i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.658117478 + 0.02736754357i\) |
\(L(1)\) |
\(\approx\) |
\(1.370684422 - 0.05121653558i\) |
\(L(1)\) |
\(\approx\) |
\(1.370684422 - 0.05121653558i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + iT \) |
| 13 | \( 1 + (0.866 + 0.5i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.866 + 0.5i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.866 - 0.5i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (-0.866 - 0.5i)T \) |
| 59 | \( 1 + (-0.866 + 0.5i)T \) |
| 61 | \( 1 + (-0.866 - 0.5i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−25.22432489588249970895356788768, −24.415449282206169601227426693375, −23.80391379084602788253831270610, −22.60966140695875357516535931200, −21.39598098534829476162424939676, −20.883277703948023434344225375584, −19.965649169976605058167389134557, −19.25114798326689564629202903155, −18.24085850094605387389968663679, −16.984282454029188605452472010781, −15.94316520578482833021136372418, −15.38897450685749156267614271067, −14.357469375158336925282577516234, −13.53340976735891431143199711864, −12.419480039766621521962801835401, −11.1580778029597893685663494469, −10.57818276948555516434983197726, −9.00256577564526329217478094810, −8.26226193544593821092906960778, −7.86695897453857629477333979848, −6.08620439907797318231562167499, −4.67424360842209317568712633495, −3.99908487274460559695015635207, −2.82694440992677325210044810674, −1.26609937160016274050044963239,
1.42806572618333999370289658856, 2.58179767435972632377288653572, 3.82930592667673492994427049364, 4.74036559492567798180488071290, 6.58078125044898069959576808515, 7.389498151969171849068003164450, 8.183909522945752272014178440690, 9.09338950521050211046582128157, 10.40433307986318573909019344636, 11.609276419782717829625422007563, 12.18276980517851562744763092588, 13.634864348988892917600061302021, 14.21983027275196210390953145270, 15.288153689631662727698965094252, 15.75504719544552426480374974212, 17.47844867939691005613812546250, 18.2462010233113721899476179719, 18.942961479746232491628043319095, 20.05163513603811801075495611574, 20.53464062555134935002880424703, 21.5802062571797331375671482335, 22.896498723620148722547351523362, 23.637939133089488034013864535129, 24.34526996665123663048002322864, 25.41918897215654124069428139672