L(s) = 1 | + (−0.663 − 0.748i)2-s + (0.354 + 0.935i)3-s + (−0.120 + 0.992i)4-s + (0.239 + 0.970i)5-s + (0.464 − 0.885i)6-s + (0.822 − 0.568i)8-s + (−0.748 + 0.663i)9-s + (0.568 − 0.822i)10-s + (−0.663 + 0.748i)11-s + (−0.970 + 0.239i)12-s + (−0.822 + 0.568i)15-s + (−0.970 − 0.239i)16-s + (0.568 + 0.822i)17-s + (0.992 + 0.120i)18-s − i·19-s + (−0.992 + 0.120i)20-s + ⋯ |
L(s) = 1 | + (−0.663 − 0.748i)2-s + (0.354 + 0.935i)3-s + (−0.120 + 0.992i)4-s + (0.239 + 0.970i)5-s + (0.464 − 0.885i)6-s + (0.822 − 0.568i)8-s + (−0.748 + 0.663i)9-s + (0.568 − 0.822i)10-s + (−0.663 + 0.748i)11-s + (−0.970 + 0.239i)12-s + (−0.822 + 0.568i)15-s + (−0.970 − 0.239i)16-s + (0.568 + 0.822i)17-s + (0.992 + 0.120i)18-s − i·19-s + (−0.992 + 0.120i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02605656259 + 0.5092636916i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02605656259 + 0.5092636916i\) |
\(L(1)\) |
\(\approx\) |
\(0.6496009466 + 0.2553007540i\) |
\(L(1)\) |
\(\approx\) |
\(0.6496009466 + 0.2553007540i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.663 - 0.748i)T \) |
| 3 | \( 1 + (0.354 + 0.935i)T \) |
| 5 | \( 1 + (0.239 + 0.970i)T \) |
| 11 | \( 1 + (-0.663 + 0.748i)T \) |
| 17 | \( 1 + (0.568 + 0.822i)T \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.748 + 0.663i)T \) |
| 31 | \( 1 + (0.464 - 0.885i)T \) |
| 37 | \( 1 + (0.464 - 0.885i)T \) |
| 41 | \( 1 + (-0.935 + 0.354i)T \) |
| 43 | \( 1 + (-0.885 + 0.464i)T \) |
| 47 | \( 1 + (-0.992 + 0.120i)T \) |
| 53 | \( 1 + (0.568 + 0.822i)T \) |
| 59 | \( 1 + (-0.239 - 0.970i)T \) |
| 61 | \( 1 + (-0.568 + 0.822i)T \) |
| 67 | \( 1 + (-0.992 + 0.120i)T \) |
| 71 | \( 1 + (0.935 - 0.354i)T \) |
| 73 | \( 1 + (0.663 - 0.748i)T \) |
| 79 | \( 1 + (0.120 + 0.992i)T \) |
| 83 | \( 1 + (0.935 + 0.354i)T \) |
| 89 | \( 1 + iT \) |
| 97 | \( 1 + (-0.239 + 0.970i)T \) |
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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
−20.51050732600481763002696259725, −19.98817397770538231202651246811, −19.033522460255726795728229831346, −18.47056637759490895023459892209, −17.86079790288028718404379532515, −16.873950654887811075106297888142, −16.434627078729763815548598762857, −15.569371436260178738689493482217, −14.56675240653430578505554681356, −13.706237320304295604977855590113, −13.38898306815122439895484530233, −12.22361669439753242519616616900, −11.55214424999653668393685017754, −10.21112797477571010334612387026, −9.56473077826128094730807591474, −8.50279640661703611816787829178, −8.184709377420451269031322998392, −7.41708514935380316484475574692, −6.31884936625191381169795827299, −5.67893139034078198960297246925, −4.90627307631711405688308890357, −3.44462831276698233648185152862, −2.11294936311151133577770494517, −1.29744182815588497712475376006, −0.24076046548218396697201268890,
1.845897825522741746725163807585, 2.572808899219365707009915022645, 3.39008683100331429101288978429, 4.185929295619545217790551118765, 5.221480942067085427329403342435, 6.452915127876418371183645806881, 7.59729431109380660351056772334, 8.126600018608660573189433947529, 9.29985364626676174916972124906, 9.84553267344799636083135917255, 10.52065349511546453369355449842, 11.07077872786451632891790878200, 11.96389415459663832732385753857, 13.069088628563367106922535222717, 13.781738406859368906861132840520, 14.84360246432327209068176572901, 15.338557183704274482845594591796, 16.33675049810610053732939663323, 17.10213682507975986998521545131, 17.95317612774144364277076291380, 18.50163241807474858855048862199, 19.52612650048495894660064798465, 19.985804658418881724333695788844, 20.899176739511215451283739375838, 21.49849930818541286434792049661