L(s) = 1 | + 2-s − i·3-s + 4-s − i·6-s − i·7-s + 8-s − 9-s − 11-s − i·12-s + 13-s − i·14-s + 16-s − 17-s − 18-s + i·19-s + ⋯ |
L(s) = 1 | + 2-s − i·3-s + 4-s − i·6-s − i·7-s + 8-s − 9-s − 11-s − i·12-s + 13-s − i·14-s + 16-s − 17-s − 18-s + i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.632022581 - 1.185436015i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.632022581 - 1.185436015i\) |
\(L(1)\) |
\(\approx\) |
\(1.637383524 - 0.6882919030i\) |
\(L(1)\) |
\(\approx\) |
\(1.637383524 - 0.6882919030i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 - iT \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + iT \) |
| 73 | \( 1 \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - iT \) |
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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
−27.559681584107962079962844832421, −26.112236835561584189516754382532, −25.631572264368867810164829318112, −24.406630282190555289918637489250, −23.42230632596758788179517169609, −22.42935726000173856554614282005, −21.720541548581737407516243592107, −20.94069995245874763287118440213, −20.1844398689865297481265216076, −18.86791037614366990200049363256, −17.52691690898407662370291873364, −16.12192390220646033925562660236, −15.573647329818375593848694249374, −14.90706103716289258319360564935, −13.60364854937547757163733857495, −12.71296511855203266759140036694, −11.27789214106169524768839475049, −10.86047071103146404180716066841, −9.327587326901655809914791173959, −8.253435513622746741880674155282, −6.558753569757280301894658723697, −5.44158136636232032508847350865, −4.69754931366315092298568661469, −3.32250613779158287304168052522, −2.36880494449159151423437170477,
1.34997355114832335988066479918, 2.72922698673276380900083951834, 3.99224757851488732060259449770, 5.40352823974154834541122902594, 6.50960060648362454782247058557, 7.3821224404951073427123883112, 8.39756680658782042214861647663, 10.47289305057252004445979440103, 11.21549415102181018204792272383, 12.4378802915916927408865661539, 13.41963422183659660208322268419, 13.74388001199481144310911494500, 15.0664872964205282579052344200, 16.213694703124230435087629396940, 17.233260430279082945514532374642, 18.39796226043886493051670646172, 19.485956103063139823791063808590, 20.44507564083346419484082003441, 21.119470578354510170076228194458, 22.703838733315383532633930699782, 23.251448079165258814554635580952, 23.92499883785298600979048103892, 24.91315579314334377445084850999, 25.770170676927291264873408825, 26.75072038736064020680842857593