L(s) = 1 | + (−0.755 + 0.654i)2-s + (0.540 − 0.841i)3-s + (0.142 − 0.989i)4-s + (0.415 − 0.909i)5-s + (0.142 + 0.989i)6-s + (0.959 + 0.281i)7-s + (0.540 + 0.841i)8-s + (−0.415 − 0.909i)9-s + (0.281 + 0.959i)10-s + (−0.755 − 0.654i)11-s + (−0.755 − 0.654i)12-s + (0.959 − 0.281i)13-s + (−0.909 + 0.415i)14-s + (−0.540 − 0.841i)15-s + (−0.959 − 0.281i)16-s + (0.989 − 0.142i)17-s + ⋯ |
L(s) = 1 | + (−0.755 + 0.654i)2-s + (0.540 − 0.841i)3-s + (0.142 − 0.989i)4-s + (0.415 − 0.909i)5-s + (0.142 + 0.989i)6-s + (0.959 + 0.281i)7-s + (0.540 + 0.841i)8-s + (−0.415 − 0.909i)9-s + (0.281 + 0.959i)10-s + (−0.755 − 0.654i)11-s + (−0.755 − 0.654i)12-s + (0.959 − 0.281i)13-s + (−0.909 + 0.415i)14-s + (−0.540 − 0.841i)15-s + (−0.959 − 0.281i)16-s + (0.989 − 0.142i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0819 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0819 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9870031702 - 0.9091569914i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9870031702 - 0.9091569914i\) |
\(L(1)\) |
\(\approx\) |
\(0.9824363307 - 0.3238827201i\) |
\(L(1)\) |
\(\approx\) |
\(0.9824363307 - 0.3238827201i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.755 + 0.654i)T \) |
| 3 | \( 1 + (0.540 - 0.841i)T \) |
| 5 | \( 1 + (0.415 - 0.909i)T \) |
| 7 | \( 1 + (0.959 + 0.281i)T \) |
| 11 | \( 1 + (-0.755 - 0.654i)T \) |
| 13 | \( 1 + (0.959 - 0.281i)T \) |
| 17 | \( 1 + (0.989 - 0.142i)T \) |
| 19 | \( 1 + (-0.989 - 0.142i)T \) |
| 31 | \( 1 + (-0.540 - 0.841i)T \) |
| 37 | \( 1 + (0.909 - 0.415i)T \) |
| 41 | \( 1 + (0.909 + 0.415i)T \) |
| 43 | \( 1 + (-0.540 + 0.841i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (0.959 + 0.281i)T \) |
| 59 | \( 1 + (-0.959 + 0.281i)T \) |
| 61 | \( 1 + (0.540 + 0.841i)T \) |
| 67 | \( 1 + (-0.654 - 0.755i)T \) |
| 71 | \( 1 + (0.654 + 0.755i)T \) |
| 73 | \( 1 + (-0.989 - 0.142i)T \) |
| 79 | \( 1 + (-0.281 - 0.959i)T \) |
| 83 | \( 1 + (-0.415 - 0.909i)T \) |
| 89 | \( 1 + (-0.540 + 0.841i)T \) |
| 97 | \( 1 + (-0.909 - 0.415i)T \) |
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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
−22.82074360383724429668126779157, −21.609266371115014912033816133867, −21.28313566096116974854387342233, −20.65783151333890188192372528246, −19.83246846663934897899330134386, −18.75921767751761136576323721483, −18.26526372462631427561490314553, −17.334553033546137357535535245829, −16.56328276867922264642240333188, −15.5149825661910145796613392029, −14.72260897023252548918841995055, −13.91469273361802349491794104971, −13.01301858092881240592565853106, −11.70667463741280134127196276745, −10.72841023776329965508570568222, −10.50462782395582199997855575462, −9.616240502381381798192440667907, −8.56373267439556824721300259948, −7.89644137613795026949073079249, −6.97912553000638186870052078952, −5.50406310897121630242265600271, −4.27855744894619631127689804824, −3.470957445491784709538572317116, −2.41963435744375441865875756822, −1.64733796043396115877275923758,
0.82941012969308328708024548070, 1.62092598275826465145818794181, 2.66914907335436209287233926233, 4.421586394575647460747661007609, 5.73329957125177618805901123875, 5.98422757578736263092497812753, 7.53595438704848417736199003034, 8.13506545386127772471402117029, 8.667743615469465203253024540, 9.485656385406576370948573500107, 10.72212112233345369818376685388, 11.59016721501784827328918660679, 12.81475276217498104905917902869, 13.49757291148857219097168720536, 14.36802517473109321202785033095, 15.10424460686506165118267475371, 16.15756922256613137143806163730, 16.9168183369827128550442382899, 17.859191812524136636105373727529, 18.32524898921016397021348636418, 19.08265664695728291358104986909, 20.05626117882364675092230689266, 20.84920716555099268615715836621, 21.329248667574887653939827421251, 23.21469580951098980415364264160