L(s) = 1 | − 1.41·2-s + 1.73i·3-s + 2.00·4-s − 2.44i·6-s + (6.73 + 1.91i)7-s − 2.82·8-s − 2.99·9-s + 17.5·11-s + 3.46i·12-s + 4.83i·13-s + (−9.52 − 2.70i)14-s + 4.00·16-s + 18.0i·17-s + 4.24·18-s + 9.13i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.500·4-s − 0.408i·6-s + (0.961 + 0.273i)7-s − 0.353·8-s − 0.333·9-s + 1.59·11-s + 0.288i·12-s + 0.371i·13-s + (−0.680 − 0.193i)14-s + 0.250·16-s + 1.06i·17-s + 0.235·18-s + 0.480i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.676851591\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.676851591\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-6.73 - 1.91i)T \) |
good | 11 | \( 1 - 17.5T + 121T^{2} \) |
| 13 | \( 1 - 4.83iT - 169T^{2} \) |
| 17 | \( 1 - 18.0iT - 289T^{2} \) |
| 19 | \( 1 - 9.13iT - 361T^{2} \) |
| 23 | \( 1 + 3.72T + 529T^{2} \) |
| 29 | \( 1 + 1.12T + 841T^{2} \) |
| 31 | \( 1 + 57.0iT - 961T^{2} \) |
| 37 | \( 1 - 41.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 11.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 64.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 77.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 77.5T + 2.80e3T^{2} \) |
| 59 | \( 1 + 87.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 5.36iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 47.1T + 4.48e3T^{2} \) |
| 71 | \( 1 + 58.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 53.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 74.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 28.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 107. iT - 9.40e3T^{2} \) |
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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
−9.531130870382445197611205164365, −9.319053499009460982664018854214, −8.264914759065952140578414420227, −7.72421662165810971502907773813, −6.41287870898828784525771454928, −5.83707676288051258020312452917, −4.45298066502602274634597169970, −3.80853279777728010555889868464, −2.24051629699359034770510634399, −1.19985585380339539797477139174,
0.78585985437312698492956363601, 1.64433743770405973901230478588, 2.92453049398599917038098967747, 4.25016802189819331445289775644, 5.33741090630526726406216840534, 6.47840004857332945736610883483, 7.14239341950295697103841660758, 7.86813151674538662708122855410, 8.791541623565028149212523763263, 9.299521400158278696493398309480