L(s) = 1 | − 3.53i·2-s − 3i·3-s − 4.46·4-s − 10.5·6-s − 7i·7-s − 12.4i·8-s − 9·9-s − 2.93·11-s + 13.4i·12-s + 19.0i·13-s − 24.7·14-s − 79.7·16-s + 122. i·17-s + 31.7i·18-s − 107.·19-s + ⋯ |
L(s) = 1 | − 1.24i·2-s − 0.577i·3-s − 0.558·4-s − 0.720·6-s − 0.377i·7-s − 0.551i·8-s − 0.333·9-s − 0.0805·11-s + 0.322i·12-s + 0.406i·13-s − 0.471·14-s − 1.24·16-s + 1.74i·17-s + 0.416i·18-s − 1.29·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2748148176\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2748148176\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7iT \) |
good | 2 | \( 1 + 3.53iT - 8T^{2} \) |
| 11 | \( 1 + 2.93T + 1.33e3T^{2} \) |
| 13 | \( 1 - 19.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 122. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 107.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 210. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 95.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 97.1iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 491.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 43.0iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 473. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 183. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 760.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 198.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 309. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 665.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 621. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 24.7T + 4.93e5T^{2} \) |
| 83 | \( 1 - 406. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 261.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.00e3iT - 9.12e5T^{2} \) |
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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
−10.13267327480511023688443053954, −8.884484148176700563313969444454, −8.141235261503905465757917128786, −6.81884674080411514778326676708, −6.21815021159355175710758615853, −4.52290368930383097628961529034, −3.64905251990573294445280500922, −2.34224726254288258447271356778, −1.50147145185159353282173626839, −0.07812670829720184653250556573,
2.25852779927536283359697924654, 3.65464113277708188772499867254, 5.12339337988887692799664641898, 5.47646647078565872685525504207, 6.70563780339687115985473973071, 7.46468960638725468710410316764, 8.444791377098424965467956167625, 9.177946855711492767613530452866, 10.08166209625501934692489172243, 11.25973190252054339547365832034