L(s) = 1 | − 3.56i·2-s − 3i·3-s − 4.68·4-s − 10.6·6-s + 7i·7-s − 11.8i·8-s − 9·9-s − 5.19·11-s + 14.0i·12-s + 54.5i·13-s + 24.9·14-s − 79.5·16-s − 16.1i·17-s + 32.0i·18-s − 87.4·19-s + ⋯ |
L(s) = 1 | − 1.25i·2-s − 0.577i·3-s − 0.585·4-s − 0.726·6-s + 0.377i·7-s − 0.521i·8-s − 0.333·9-s − 0.142·11-s + 0.338i·12-s + 1.16i·13-s + 0.475·14-s − 1.24·16-s − 0.230i·17-s + 0.419i·18-s − 1.05·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6224279365\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6224279365\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.56iT - 8T^{2} \) |
| 11 | \( 1 + 5.19T + 1.33e3T^{2} \) |
| 13 | \( 1 - 54.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 16.1iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 87.4T + 6.85e3T^{2} \) |
| 23 | \( 1 - 176. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 142.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 94.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 17.3iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 210.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 521. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 105. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 108. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 210.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 324. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 793.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 315. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 425.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 283. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 843.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.53e3iT - 9.12e5T^{2} \) |
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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
−10.90168445696783595060228021377, −9.557507129208766988154447368812, −9.169129359276149759340284120216, −7.86035935205589560413627603258, −6.88395025011908049600612009023, −5.93263983812201414376527455143, −4.52524891133272392457023782904, −3.41908978691637535910624205904, −2.25025037778850724628986485304, −1.45469342846911195459752545223,
0.17998808093967275412602468942, 2.41124529685428203914162521764, 3.89311467075332569135161413723, 4.95847608809309876419895816283, 5.81761910568531620050127098119, 6.67812544748660144602872388789, 7.69504306666785861827085306678, 8.395386134598182602306056711834, 9.230231509477348081312715052503, 10.61680173897015524626678623712