| L(s) = 1 | − 1.41·2-s − 5.41·3-s + 2.00·4-s − 2.23i·5-s + 7.66·6-s + 8.24i·7-s − 2.82·8-s + 20.3·9-s + 3.16i·10-s + 15.8i·11-s − 10.8·12-s − 14.3·13-s − 11.6i·14-s + 12.1i·15-s + 4.00·16-s − 10.1i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.80·3-s + 0.500·4-s − 0.447i·5-s + 1.27·6-s + 1.17i·7-s − 0.353·8-s + 2.26·9-s + 0.316i·10-s + 1.43i·11-s − 0.903·12-s − 1.10·13-s − 0.832i·14-s + 0.807i·15-s + 0.250·16-s − 0.598i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.147545 - 0.191337i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.147545 - 0.191337i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-22.2 - 5.84i)T \) |
| good | 3 | \( 1 + 5.41T + 9T^{2} \) |
| 7 | \( 1 - 8.24iT - 49T^{2} \) |
| 11 | \( 1 - 15.8iT - 121T^{2} \) |
| 13 | \( 1 + 14.3T + 169T^{2} \) |
| 17 | \( 1 + 10.1iT - 289T^{2} \) |
| 19 | \( 1 + 36.5iT - 361T^{2} \) |
| 29 | \( 1 - 6.46T + 841T^{2} \) |
| 31 | \( 1 + 42.8T + 961T^{2} \) |
| 37 | \( 1 + 63.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 37.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 6.00iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 32.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 36.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 6.65T + 3.48e3T^{2} \) |
| 61 | \( 1 + 55.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.45iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 118.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 82.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 133. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 67.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 104. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 98.6iT - 9.40e3T^{2} \) |
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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
−11.63853423388903362563214539813, −10.91135040749510304398437959967, −9.667379533366011273644813280258, −9.170199438784641790292162797851, −7.35252435503514435310127118808, −6.74739336182501520206121853151, −5.27130891075658291907434332896, −4.89362661694179991162623717106, −2.12504045406519273001646631012, −0.23738347240363406452066612275,
1.16022260764845045917918850914, 3.71160434675757575011063568814, 5.25443053973467670280118081200, 6.28638391102918740656331753402, 7.06087316028652854249305902183, 8.096445166921115076403163349273, 9.864662745591925308718345913450, 10.50724161776968928408523932955, 11.05145855662610522961346565798, 11.92505901501058652698454317732
