L(s) = 1 | + 1.41i·2-s − 2.97·3-s − 2.00·4-s − 4.20i·6-s + (−5.43 + 4.41i)7-s − 2.82i·8-s − 0.171·9-s + 9.24·11-s + 5.94·12-s − 15.5·13-s + (−6.24 − 7.68i)14-s + 4.00·16-s + 12.9·17-s − 0.242i·18-s − 23.7i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.990·3-s − 0.500·4-s − 0.700i·6-s + (−0.776 + 0.630i)7-s − 0.353i·8-s − 0.0190·9-s + 0.840·11-s + 0.495·12-s − 1.19·13-s + (−0.445 − 0.548i)14-s + 0.250·16-s + 0.759·17-s − 0.0134i·18-s − 1.25i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 + 0.412i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.911 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.631330 - 0.136154i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.631330 - 0.136154i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (5.43 - 4.41i)T \) |
good | 3 | \( 1 + 2.97T + 9T^{2} \) |
| 11 | \( 1 - 9.24T + 121T^{2} \) |
| 13 | \( 1 + 15.5T + 169T^{2} \) |
| 17 | \( 1 - 12.9T + 289T^{2} \) |
| 19 | \( 1 + 23.7iT - 361T^{2} \) |
| 23 | \( 1 + 7.58iT - 529T^{2} \) |
| 29 | \( 1 - 43.6T + 841T^{2} \) |
| 31 | \( 1 + 35.1iT - 961T^{2} \) |
| 37 | \( 1 - 4.11iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 26.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 11.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 16.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 49.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 69.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 83.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 101. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 34.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 111.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 128.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 159.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 13.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 109.T + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.42071305273671186288509863572, −10.10887707059987509262194083013, −9.392631271349718913571554683020, −8.389675890081233185132035496932, −7.03573926962895135790132685680, −6.37437569573883204094693497409, −5.47848251469750660062456805925, −4.54770452498404460558423647481, −2.85526436324135648624003803809, −0.41320211784364477584626060048,
1.08281107601509288331751789794, 3.01530799211414764916574534285, 4.21421098803094297114030648173, 5.38680384820372155617578066691, 6.38468670700563899876345080104, 7.40858800867321652548847325629, 8.762498663501532747719556784669, 9.984384474600555589131078374718, 10.29070191397558610307730722249, 11.43539701820029506895306468695