L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + i·7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)11-s + (−0.951 − 0.309i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.587 − 0.809i)22-s + (−0.951 + 0.309i)23-s − 26-s + (0.587 + 0.809i)28-s + (−0.809 + 0.587i)29-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + i·7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)11-s + (−0.951 − 0.309i)13-s + (0.309 + 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.587 + 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.587 − 0.809i)22-s + (−0.951 + 0.309i)23-s − 26-s + (0.587 + 0.809i)28-s + (−0.809 + 0.587i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.500517376 - 0.3354053717i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.500517376 - 0.3354053717i\) |
\(L(1)\) |
\(\approx\) |
\(1.568325357 - 0.2637720242i\) |
\(L(1)\) |
\(\approx\) |
\(1.568325357 - 0.2637720242i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (-0.309 - 0.951i)T \) |
| 13 | \( 1 + (-0.951 - 0.309i)T \) |
| 17 | \( 1 + (-0.587 + 0.809i)T \) |
| 19 | \( 1 + (0.809 + 0.587i)T \) |
| 23 | \( 1 + (-0.951 + 0.309i)T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.809 - 0.587i)T \) |
| 37 | \( 1 + (0.951 + 0.309i)T \) |
| 41 | \( 1 + (-0.309 + 0.951i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.587 + 0.809i)T \) |
| 53 | \( 1 + (-0.587 - 0.809i)T \) |
| 59 | \( 1 + (0.309 - 0.951i)T \) |
| 61 | \( 1 + (0.309 + 0.951i)T \) |
| 67 | \( 1 + (0.587 - 0.809i)T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.951 - 0.309i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
| 83 | \( 1 + (0.587 - 0.809i)T \) |
| 89 | \( 1 + (0.309 + 0.951i)T \) |
| 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
−31.49021734242127749097715920059, −30.55175148281726817928042109858, −29.58241656806275935505448549415, −28.583548396427199769802884959412, −26.83509033747514243041378817791, −26.02986409408344734360583003054, −24.73064089440329404969479777403, −23.82125164417064322726257535889, −22.80441985600756051378580499354, −21.909146499077764059741811978454, −20.45067488504524048542420376773, −19.92076112186553457926712270858, −17.902962447867428201660514618846, −16.81098876813299198004577070645, −15.70488739237512564476279692926, −14.48037543471664478420051856992, −13.528387203519761674367498103075, −12.367114658649210502218074326086, −11.13915626427146169858320328057, −9.70608496656119102617946307698, −7.622888775403995865228620534878, −6.898262392985003517434294698198, −5.11948836465730031999563450626, −4.07164998331708953342475664812, −2.346783335822827705966530568462,
2.13003584549835795485031888699, 3.4990665827546994008466567251, 5.21158575009420122640494412625, 6.13425545853426539636435369620, 7.88488524950713748321538314315, 9.598685085517007843492527326539, 11.02264667588094260024054059230, 12.10896445569681296201747886452, 13.12466705398785078645358834634, 14.43880980150086408171908887480, 15.403331356096559521499977589526, 16.518839937767002240198917284961, 18.302735615967003015561630796453, 19.38726332056656576314937507763, 20.50788704266673045546670993521, 21.89894366997041236520671893601, 22.15415500748073529763906485034, 23.85651876209231772884189300550, 24.50213092292981182765683458704, 25.650415157416032969790826720534, 27.17761442626570509790979599506, 28.473244001471756911057561638071, 29.25757391613924619160652125092, 30.36227471728137211930436141970, 31.55554150856976834658471136611