L(s) = 1 | + 6.38i·2-s − 24.7·4-s + 16.4i·5-s + 84.4·7-s − 55.9i·8-s − 104.·10-s − 31.3i·11-s + 43.8·13-s + 539. i·14-s − 39.1·16-s − 243. i·17-s + 602.·19-s − 406. i·20-s + 199.·22-s − 401. i·23-s + ⋯ |
L(s) = 1 | + 1.59i·2-s − 1.54·4-s + 0.657i·5-s + 1.72·7-s − 0.873i·8-s − 1.04·10-s − 0.258i·11-s + 0.259·13-s + 2.75i·14-s − 0.153·16-s − 0.841i·17-s + 1.66·19-s − 1.01i·20-s + 0.413·22-s − 0.758i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.707762598\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.707762598\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 + 453. iT \) |
good | 2 | \( 1 - 6.38iT - 16T^{2} \) |
| 5 | \( 1 - 16.4iT - 625T^{2} \) |
| 7 | \( 1 - 84.4T + 2.40e3T^{2} \) |
| 11 | \( 1 + 31.3iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 43.8T + 2.85e4T^{2} \) |
| 17 | \( 1 + 243. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 602.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 401. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 1.57e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 306.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 2.50e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.68e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.97e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 890. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 4.49e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 + 4.99e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 5.52e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 6.51e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.30e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 7.10e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.73e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 9.38e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.68e4T + 8.85e7T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.72593537230354649189478607192, −9.255438444660924578025305824371, −8.572193723426590795536468400875, −7.51955487153273140497968655249, −7.29821549107505314368821194294, −6.03362328972085207443281175284, −5.18474400911905035929013051073, −4.46099406095042747914890085439, −2.79950733898324913991985103198, −1.06961512223574647406646129808,
0.950647312262978150945810779572, 1.52207302755826796076918710834, 2.69752383363375753682551473305, 4.11639884053370584557116128100, 4.70942095220942345055938458553, 5.77095364532825272804690477295, 7.60925796938174672858367774902, 8.281689342722499273272026052395, 9.293516777362704605559816393339, 9.982889688952851980906867179855