L(s) = 1 | + (0.974 + 0.222i)3-s + (0.900 + 0.433i)9-s + (0.900 − 0.433i)11-s + (−0.433 − 0.900i)13-s + (0.781 − 0.623i)17-s + 19-s + (−0.781 − 0.623i)23-s + (0.781 + 0.623i)27-s + (−0.623 − 0.781i)29-s − 31-s + (0.974 − 0.222i)33-s + (0.781 − 0.623i)37-s + (−0.222 − 0.974i)39-s + (−0.222 + 0.974i)41-s + (−0.974 + 0.222i)43-s + ⋯ |
L(s) = 1 | + (0.974 + 0.222i)3-s + (0.900 + 0.433i)9-s + (0.900 − 0.433i)11-s + (−0.433 − 0.900i)13-s + (0.781 − 0.623i)17-s + 19-s + (−0.781 − 0.623i)23-s + (0.781 + 0.623i)27-s + (−0.623 − 0.781i)29-s − 31-s + (0.974 − 0.222i)33-s + (0.781 − 0.623i)37-s + (−0.222 − 0.974i)39-s + (−0.222 + 0.974i)41-s + (−0.974 + 0.222i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.270809965 - 0.4146634632i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.270809965 - 0.4146634632i\) |
\(L(1)\) |
\(\approx\) |
\(1.570295219 - 0.07483347803i\) |
\(L(1)\) |
\(\approx\) |
\(1.570295219 - 0.07483347803i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.974 + 0.222i)T \) |
| 11 | \( 1 + (0.900 - 0.433i)T \) |
| 13 | \( 1 + (-0.433 - 0.900i)T \) |
| 17 | \( 1 + (0.781 - 0.623i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.781 - 0.623i)T \) |
| 29 | \( 1 + (-0.623 - 0.781i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.781 - 0.623i)T \) |
| 41 | \( 1 + (-0.222 + 0.974i)T \) |
| 43 | \( 1 + (-0.974 + 0.222i)T \) |
| 47 | \( 1 + (-0.433 - 0.900i)T \) |
| 53 | \( 1 + (0.781 + 0.623i)T \) |
| 59 | \( 1 + (-0.222 - 0.974i)T \) |
| 61 | \( 1 + (0.623 + 0.781i)T \) |
| 67 | \( 1 - iT \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (0.433 - 0.900i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (-0.433 + 0.900i)T \) |
| 89 | \( 1 + (0.900 + 0.433i)T \) |
| 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
−21.759429326680708911282630707404, −20.86638152096789907132001706210, −20.01912110838992773177121203901, −19.59285238532354425924880218539, −18.695453791901524292958889618329, −18.0393308173273068511481359486, −16.96580709623182496066135505446, −16.25992488643832669491213967796, −15.21221284113519877746452604392, −14.49351458964147408371742956807, −14.02960476815664409815436638823, −13.05819028279665187281066864669, −12.18867579867393423397662026065, −11.559933647722157220403041248553, −10.18658443684219413374973365978, −9.48963865046933947866112299873, −8.8955808112335931047440996797, −7.781949694943528120208610344319, −7.19860176975935382842342531020, −6.27916688285032113856758287737, −5.05502339480678777139262681373, −3.92086483436542884097028223635, −3.366483699414736465603730534342, −2.01753900695299011920160323447, −1.3957558473467530496899152211,
0.97624363601436087131166983553, 2.20504358356462793900489568080, 3.19118577599403573949971777457, 3.8545634478395658522757650180, 4.97800642992325890443989572457, 5.92667292084713283699616623359, 7.16396831838223977300124054151, 7.83241188291783552399710129281, 8.64963120930251623899627949258, 9.63221971476863080332476486762, 10.02746027685950645377842283396, 11.25539233069542049096362039665, 12.1067300271489040296294027279, 13.056291489947753496413870437665, 13.839228868599897303884643602589, 14.57751563937780883850004186978, 15.11549814365098659012055090310, 16.24009158626917082766975560102, 16.658061358309791837868177637876, 17.97386187883566835064378213781, 18.55685302104595152766089500759, 19.52750292856316616287979521351, 20.127215957605724461303543290874, 20.66060209660265729324452222397, 21.74805833024066522617991388879