| L(s) = 1 | + 1.20·3-s + 2.23i·5-s + 12.3i·7-s − 7.54·9-s + 9.41·11-s + 4.18i·13-s + 2.69i·15-s + 21.3·17-s − 0.990·19-s + 14.9i·21-s − 0.480i·23-s − 5.00·25-s − 19.9·27-s + 32.5i·29-s − 23.2i·31-s + ⋯ |
| L(s) = 1 | + 0.401·3-s + 0.447i·5-s + 1.77i·7-s − 0.838·9-s + 0.856·11-s + 0.321i·13-s + 0.179i·15-s + 1.25·17-s − 0.0521·19-s + 0.711i·21-s − 0.0208i·23-s − 0.200·25-s − 0.738·27-s + 1.12i·29-s − 0.750i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 - 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.354 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.28392 + 0.885867i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28392 + 0.885867i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| good | 3 | \( 1 - 1.20T + 9T^{2} \) |
| 7 | \( 1 - 12.3iT - 49T^{2} \) |
| 11 | \( 1 - 9.41T + 121T^{2} \) |
| 13 | \( 1 - 4.18iT - 169T^{2} \) |
| 17 | \( 1 - 21.3T + 289T^{2} \) |
| 19 | \( 1 + 0.990T + 361T^{2} \) |
| 23 | \( 1 + 0.480iT - 529T^{2} \) |
| 29 | \( 1 - 32.5iT - 841T^{2} \) |
| 31 | \( 1 + 23.2iT - 961T^{2} \) |
| 37 | \( 1 + 55.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 16.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 57.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 17.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 26.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 76.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 104. iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 2.30T + 4.48e3T^{2} \) |
| 71 | \( 1 - 54.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 73.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 39.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 58.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 28.3T + 7.92e3T^{2} \) |
| 97 | \( 1 + 4.84T + 9.40e3T^{2} \) |
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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
−12.61388457849602592245324543691, −11.90988009117504654533552214475, −11.03821460361975061352259620326, −9.450061453533533912581062779918, −8.914047458095960224868203984309, −7.79048895517079733768870504671, −6.26172603866793196704881865931, −5.41531118332341234502383589492, −3.45418945040249262643581307525, −2.25236877694095857072075680715,
1.02077279725449790770897169458, 3.31631728810382447118053103570, 4.43946122418911770148565771052, 6.01024433119520573289783555711, 7.38029266514603228516674531043, 8.215076435185103588110391161592, 9.443726957413767556591724823750, 10.38125966365550690197887330324, 11.44751219931856095070436979160, 12.50912336072405491544018823632
