L(s) = 1 | + (0.623 − 0.781i)2-s + (−0.900 − 0.433i)3-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)5-s + (−0.900 + 0.433i)6-s + (−0.900 − 0.433i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + (0.623 − 0.781i)11-s + (−0.222 + 0.974i)12-s + (0.623 − 0.781i)13-s + (0.623 + 0.781i)15-s + (−0.900 + 0.433i)16-s + (−0.222 + 0.974i)17-s + 18-s + 19-s + ⋯ |
L(s) = 1 | + (0.623 − 0.781i)2-s + (−0.900 − 0.433i)3-s + (−0.222 − 0.974i)4-s + (−0.900 − 0.433i)5-s + (−0.900 + 0.433i)6-s + (−0.900 − 0.433i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + (0.623 − 0.781i)11-s + (−0.222 + 0.974i)12-s + (0.623 − 0.781i)13-s + (0.623 + 0.781i)15-s + (−0.900 + 0.433i)16-s + (−0.222 + 0.974i)17-s + 18-s + 19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3084016739 - 0.6966847535i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3084016739 - 0.6966847535i\) |
\(L(1)\) |
\(\approx\) |
\(0.6673565200 - 0.6279864890i\) |
\(L(1)\) |
\(\approx\) |
\(0.6673565200 - 0.6279864890i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
good | 2 | \( 1 + (0.623 - 0.781i)T \) |
| 3 | \( 1 + (-0.900 - 0.433i)T \) |
| 5 | \( 1 + (-0.900 - 0.433i)T \) |
| 11 | \( 1 + (0.623 - 0.781i)T \) |
| 13 | \( 1 + (0.623 - 0.781i)T \) |
| 17 | \( 1 + (-0.222 + 0.974i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.222 - 0.974i)T \) |
| 29 | \( 1 + (-0.222 + 0.974i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.222 + 0.974i)T \) |
| 41 | \( 1 + (-0.900 - 0.433i)T \) |
| 43 | \( 1 + (-0.900 + 0.433i)T \) |
| 47 | \( 1 + (0.623 - 0.781i)T \) |
| 53 | \( 1 + (-0.222 - 0.974i)T \) |
| 59 | \( 1 + (-0.900 + 0.433i)T \) |
| 61 | \( 1 + (-0.222 + 0.974i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.623 + 0.781i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.623 + 0.781i)T \) |
| 89 | \( 1 + (0.623 + 0.781i)T \) |
| 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
−33.84860728268130797567468716364, −33.4980867210934649201770941163, −32.0956471197246953989057623712, −30.97347201406057966729790640252, −29.95184959307023862219221017367, −28.32694216474670484248322154627, −27.14337686086795957218035752676, −26.30410179679758554246256323106, −24.73017093373225737163692720289, −23.43547513203816601438432100477, −22.8358676312552065233982276615, −21.85059889447489981101086826274, −20.43997556185653167777783322780, −18.486531230616380999517181919975, −17.299186926816323441417437123668, −16.021899096444041578191448182, −15.35285302914697572706954057299, −13.9030365617759646104405128266, −12.06750049661719219145538821757, −11.43939016912632378456591074570, −9.40592740050215438917767077792, −7.46416823385243877355683560175, −6.41813870650314258742355562036, −4.78077931061807683666236754315, −3.67103236213400625148587958008,
1.10243920634589678238791755155, 3.59021991559673552754788675845, 5.07778570447666233172176451792, 6.43699204361261560893900551562, 8.42221009240629092763690514606, 10.488999912389959454635374465411, 11.53897447740747797473114158446, 12.4345250357034266273340176781, 13.58282255614661952312209072955, 15.336150675731618719359333403939, 16.602729950860218375770954336947, 18.2515969252552190770872343907, 19.3157260086256144632086963924, 20.40135811763955304356491623265, 21.94942948652353453437163745056, 22.822612007233742848441980031426, 23.88846337712835952973659075678, 24.621526593735992455521018558704, 27.061499162392013220642195518484, 27.99680404912355304610707158426, 28.8155034193021144430942922568, 30.11667416769505129577758486392, 30.79743730706022787391175372554, 32.236018278314702227574836393, 33.11484175795562833495853478095