L(s) = 1 | + (0.195 − 0.980i)2-s + (−0.923 − 0.384i)4-s + (0.906 + 0.421i)5-s + (−0.994 + 0.108i)7-s + (−0.557 + 0.830i)8-s + (0.591 − 0.806i)10-s + (−0.476 − 0.879i)11-s + (−0.115 − 0.993i)13-s + (−0.0882 + 0.996i)14-s + (0.704 + 0.709i)16-s + (0.415 − 0.909i)17-s + (−0.794 + 0.607i)19-s + (−0.675 − 0.737i)20-s + (−0.955 + 0.294i)22-s + (0.644 + 0.764i)25-s + (−0.996 − 0.0815i)26-s + ⋯ |
L(s) = 1 | + (0.195 − 0.980i)2-s + (−0.923 − 0.384i)4-s + (0.906 + 0.421i)5-s + (−0.994 + 0.108i)7-s + (−0.557 + 0.830i)8-s + (0.591 − 0.806i)10-s + (−0.476 − 0.879i)11-s + (−0.115 − 0.993i)13-s + (−0.0882 + 0.996i)14-s + (0.704 + 0.709i)16-s + (0.415 − 0.909i)17-s + (−0.794 + 0.607i)19-s + (−0.675 − 0.737i)20-s + (−0.955 + 0.294i)22-s + (0.644 + 0.764i)25-s + (−0.996 − 0.0815i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2110157392 - 0.2667878317i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2110157392 - 0.2667878317i\) |
\(L(1)\) |
\(\approx\) |
\(0.7332740014 - 0.5087083151i\) |
\(L(1)\) |
\(\approx\) |
\(0.7332740014 - 0.5087083151i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.195 - 0.980i)T \) |
| 5 | \( 1 + (0.906 + 0.421i)T \) |
| 7 | \( 1 + (-0.994 + 0.108i)T \) |
| 11 | \( 1 + (-0.476 - 0.879i)T \) |
| 13 | \( 1 + (-0.115 - 0.993i)T \) |
| 17 | \( 1 + (0.415 - 0.909i)T \) |
| 19 | \( 1 + (-0.794 + 0.607i)T \) |
| 31 | \( 1 + (0.288 - 0.957i)T \) |
| 37 | \( 1 + (-0.986 - 0.162i)T \) |
| 41 | \( 1 + (-0.235 - 0.971i)T \) |
| 43 | \( 1 + (-0.288 - 0.957i)T \) |
| 47 | \( 1 + (-0.0747 + 0.997i)T \) |
| 53 | \( 1 + (0.182 + 0.983i)T \) |
| 59 | \( 1 + (0.327 - 0.945i)T \) |
| 61 | \( 1 + (-0.777 + 0.628i)T \) |
| 67 | \( 1 + (0.476 - 0.879i)T \) |
| 71 | \( 1 + (0.992 + 0.122i)T \) |
| 73 | \( 1 + (-0.452 + 0.891i)T \) |
| 79 | \( 1 + (0.966 + 0.255i)T \) |
| 83 | \( 1 + (-0.999 - 0.0135i)T \) |
| 89 | \( 1 + (-0.862 - 0.505i)T \) |
| 97 | \( 1 + (0.833 + 0.552i)T \) |
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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
−17.92127980882546710995575101253, −17.41340976707784371708932007077, −16.69037515541958788441289067162, −16.433830698409620446592830391322, −15.557722055385420587841761417, −14.97119368281922611900812777134, −14.28266168244234016282707238907, −13.57125615030454164876696175324, −13.042011136228620443077032122180, −12.58180325611607295067356171937, −11.94016564557591251482032022300, −10.59014207236895669726716504236, −9.969490242549097414860585648240, −9.52346502446648618455854775556, −8.76719017303667811991685200693, −8.24406092322904516738789154141, −7.153549083760332844809095980137, −6.6430469129647041766558606552, −6.2088178036479035748248776305, −5.297750636339770490304184026023, −4.73773979391415021071665550209, −4.021798466761440312905494944061, −3.12606294682908203717535597265, −2.20044321671203055512166666407, −1.30190398263701835733007030496,
0.08751626659078968835138025402, 0.97244220080250420168455235317, 2.09635003626579831233533076903, 2.71994668641778708898352781098, 3.23627008718647958502903040571, 3.91368648437557041470435138269, 5.10852886727198138746625955152, 5.66049540197298934817579308610, 6.10562730410247621372704477679, 7.04905004986942501424952852799, 8.04611883648405447552736046009, 8.79203414236029841509183509555, 9.51660388656007014214679955895, 10.04420263349506682379883270039, 10.597362857542315530356016904660, 11.09355514745238088007887421377, 12.170196058907365948246662718141, 12.59993560436290273096706368303, 13.33881854552193808949498981476, 13.75742736276235239560109370836, 14.338774064996852012406583231554, 15.24422482613426087048509290327, 15.81615918865477789815069222127, 16.88354973481154180116039214124, 17.2442281659710643760004690535