L(s) = 1 | + (−0.759 + 0.203i)2-s + i·3-s + (−1.19 + 0.690i)4-s + (2.02 + 0.542i)5-s + (−0.203 − 0.759i)6-s + (1.65 + 2.06i)7-s + (1.88 − 1.88i)8-s − 9-s − 1.64·10-s + (−1.72 + 1.72i)11-s + (−0.690 − 1.19i)12-s + (−1.96 − 3.02i)13-s + (−1.67 − 1.22i)14-s + (−0.542 + 2.02i)15-s + (0.335 − 0.581i)16-s + (3.27 + 5.67i)17-s + ⋯ |
L(s) = 1 | + (−0.537 + 0.143i)2-s + 0.577i·3-s + (−0.598 + 0.345i)4-s + (0.906 + 0.242i)5-s + (−0.0830 − 0.310i)6-s + (0.626 + 0.779i)7-s + (0.664 − 0.664i)8-s − 0.333·9-s − 0.521·10-s + (−0.521 + 0.521i)11-s + (−0.199 − 0.345i)12-s + (−0.546 − 0.837i)13-s + (−0.448 − 0.328i)14-s + (−0.140 + 0.523i)15-s + (0.0839 − 0.145i)16-s + (0.794 + 1.37i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.517004 + 0.757849i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.517004 + 0.757849i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (-1.65 - 2.06i)T \) |
| 13 | \( 1 + (1.96 + 3.02i)T \) |
good | 2 | \( 1 + (0.759 - 0.203i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (-2.02 - 0.542i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (1.72 - 1.72i)T - 11iT^{2} \) |
| 17 | \( 1 + (-3.27 - 5.67i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.66 - 5.66i)T - 19iT^{2} \) |
| 23 | \( 1 + (-3.33 - 1.92i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.25 + 2.17i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.122 + 0.457i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (1.83 + 6.83i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-11.0 - 2.94i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.810 - 0.467i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.88 + 7.03i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.08 - 1.87i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.43 + 5.36i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + 9.39iT - 61T^{2} \) |
| 67 | \( 1 + (-8.15 - 8.15i)T + 67iT^{2} \) |
| 71 | \( 1 + (-9.44 + 2.52i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-9.79 + 2.62i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.07 - 1.86i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.52 - 1.52i)T - 83iT^{2} \) |
| 89 | \( 1 + (-6.08 + 1.62i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (4.17 + 15.5i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−12.50979155473612797870857004768, −10.83035599438047767738377379717, −10.12347534488572542196565867042, −9.461049536023252742796938967039, −8.346860145734244992331902982693, −7.75803203116162327726961125457, −5.99458325139672102412337297755, −5.17728708634806499915802100264, −3.85044805182497711785252376563, −2.14023533072444169897684304867,
0.895581836729536866729099994914, 2.35309468383612157546494743611, 4.59690272562627656397289475698, 5.39078207763221320169832312623, 6.81127366548331488025435433496, 7.81666115787639628527348428826, 8.905142542129201476400253180174, 9.564097774299052903039748547103, 10.65323796556066780950159643441, 11.32753258557882759111817365301