L(s) = 1 | + (−0.866 − 0.5i)5-s − 7-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + 23-s + (0.5 + 0.866i)25-s + (−0.866 − 0.5i)29-s + (−0.5 + 0.866i)31-s + (0.866 + 0.5i)35-s + (0.866 − 0.5i)37-s + 41-s + i·43-s + (−0.5 − 0.866i)47-s + 49-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)5-s − 7-s + (0.866 + 0.5i)11-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + 23-s + (0.5 + 0.866i)25-s + (−0.866 − 0.5i)29-s + (−0.5 + 0.866i)31-s + (0.866 + 0.5i)35-s + (0.866 − 0.5i)37-s + 41-s + i·43-s + (−0.5 − 0.866i)47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.228i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.009866751234 - 0.08504998648i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.009866751234 - 0.08504998648i\) |
\(L(1)\) |
\(\approx\) |
\(0.7489711589 + 0.01300692045i\) |
\(L(1)\) |
\(\approx\) |
\(0.7489711589 + 0.01300692045i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (-0.866 - 0.5i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + (-0.866 + 0.5i)T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.866 - 0.5i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 - T \) |
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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
−20.051820851342571628405549217150, −19.518379667225931463153303185646, −18.78008078509739720059142899071, −18.42965083046657366951572804830, −17.07124720626599415892726965726, −16.6319816599729083977472073716, −15.8327528697469999824189681965, −15.14976586838796975977172219588, −14.46866975874427349638778750429, −13.61474386406259656138433806365, −12.75472773298757003009563645368, −12.138315121843050995710480257347, −11.1054097573114446397672474917, −10.88012570085952123390486168196, −9.54202035681682666674292788793, −9.13585563622492941884262259839, −8.10633234145268597453366964290, −7.2424403326531248016505298694, −6.58489174751042521774576154999, −5.92824661687618996191598328417, −4.65576864178374651427651753006, −3.81138622093857306344715224279, −3.21697467527236805630483980411, −2.295300820881312519879963196326, −0.84426531317757688328300291137,
0.021547032819752100598319740667, 1.02994723142888675550028006623, 2.17554854796751342938794362201, 3.32623847729911933910499779534, 4.03513999909114402821890251207, 4.67973950601570400588000160043, 5.853108037364304725482503067543, 6.69931468151019569495327991320, 7.28321716243203851324599771067, 8.33839384651585990117855305370, 9.06022591910967350080113020902, 9.599380319381540550199666076416, 10.76105339142550111794896602170, 11.3363593246550607464634741626, 12.335036537058975004063270126548, 12.81948559477689613173939749125, 13.38531626418879835020091217012, 14.8132295016979200156939833167, 15.01936046813774747290912929792, 15.98321819744982247468958170661, 16.648408297108112811939613520704, 17.187887122969370622110191117761, 18.12911505019963857749120698576, 19.22440274614820209307383344523, 19.56451292478292368183529907141