L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s − 13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s + 11-s − 12-s − 13-s + 14-s + 15-s + 16-s − 17-s − 18-s + 19-s − 20-s + 21-s − 22-s + 23-s + 24-s + 25-s + 26-s − 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1723 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3915022698\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3915022698\) |
\(L(1)\) |
\(\approx\) |
\(0.3784227507\) |
\(L(1)\) |
\(\approx\) |
\(0.3784227507\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1723 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 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
−19.949608409306920623009012923893, −19.18401491155709838107346003791, −18.78150907161704501640531484776, −17.819862741887041578364785643374, −17.07385155316963319025957609122, −16.55756175368721514141698772852, −15.90473226292702786545384731260, −15.316959894080823592142367351074, −14.46797916822055020981306513076, −12.84054253315006194632061169116, −12.51331832744588268634057028763, −11.464902624892737697361695217296, −11.2889501173566626691430753316, −10.29662907118026207713618204406, −9.33349839208266606386290558348, −9.0636313064175074284897371219, −7.44123764774914389982966997655, −7.30055880498109754734223163795, −6.45067763133964313491542013499, −5.62145183963049729016120829042, −4.44056016404960703475298862343, −3.57504622099057624731221671621, −2.54193544172015522393207063447, −1.19220513823099894138286389978, −0.368909083139496750303343173473,
0.368909083139496750303343173473, 1.19220513823099894138286389978, 2.54193544172015522393207063447, 3.57504622099057624731221671621, 4.44056016404960703475298862343, 5.62145183963049729016120829042, 6.45067763133964313491542013499, 7.30055880498109754734223163795, 7.44123764774914389982966997655, 9.0636313064175074284897371219, 9.33349839208266606386290558348, 10.29662907118026207713618204406, 11.2889501173566626691430753316, 11.464902624892737697361695217296, 12.51331832744588268634057028763, 12.84054253315006194632061169116, 14.46797916822055020981306513076, 15.316959894080823592142367351074, 15.90473226292702786545384731260, 16.55756175368721514141698772852, 17.07385155316963319025957609122, 17.819862741887041578364785643374, 18.78150907161704501640531484776, 19.18401491155709838107346003791, 19.949608409306920623009012923893