| L(s) = 1 | − 1.41i·2-s − 2.00·4-s + (−4.91 + 0.905i)5-s + (−1.91 − 6.73i)7-s + 2.82i·8-s + (1.28 + 6.95i)10-s − 17.5·11-s + 4.83·13-s + (−9.52 + 2.70i)14-s + 4.00·16-s + 18.0·17-s + 9.13i·19-s + (9.83 − 1.81i)20-s + 24.7i·22-s + 3.72i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + (−0.983 + 0.181i)5-s + (−0.273 − 0.961i)7-s + 0.353i·8-s + (0.128 + 0.695i)10-s − 1.59·11-s + 0.371·13-s + (−0.680 + 0.193i)14-s + 0.250·16-s + 1.06·17-s + 0.480i·19-s + (0.491 − 0.0905i)20-s + 1.12i·22-s + 0.161i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7653096556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7653096556\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.91 - 0.905i)T \) |
| 7 | \( 1 + (1.91 + 6.73i)T \) |
| good | 11 | \( 1 + 17.5T + 121T^{2} \) |
| 13 | \( 1 - 4.83T + 169T^{2} \) |
| 17 | \( 1 - 18.0T + 289T^{2} \) |
| 19 | \( 1 - 9.13iT - 361T^{2} \) |
| 23 | \( 1 - 3.72iT - 529T^{2} \) |
| 29 | \( 1 + 1.12T + 841T^{2} \) |
| 31 | \( 1 - 57.0iT - 961T^{2} \) |
| 37 | \( 1 + 41.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 11.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 64.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 77.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 77.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 87.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 5.36iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 47.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 58.3T + 5.04e3T^{2} \) |
| 73 | \( 1 - 53.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 74.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 28.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 107.T + 9.40e3T^{2} \) |
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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
−10.65124083456569072830556585208, −9.922414961054614290362849733849, −8.671726939617327790485965013867, −7.76247962227295377041725063548, −7.26292548058769680444969512756, −5.75171872432915283082823545292, −4.64477064551224697186470192410, −3.65113137355624163134453349108, −2.85909042467767985173156862007, −1.00850000526637438436678632404,
0.35695994527119149865775597786, 2.61030286069726663370913017003, 3.76113378699834344006311908953, 5.05216539821072750922469286912, 5.65252525487066338546150914348, 6.83898700471762055815892877019, 7.914639878485065635332475785174, 8.238449998800041417842749962357, 9.283344215072061965422371590803, 10.21735101533975811175471797498
