| L(s) = 1 | + (−0.229 + 4.99i)5-s − 8.73i·7-s + 10.4i·11-s + 5.38i·13-s − 26.2·17-s + 2.70·19-s + 33.2·23-s + (−24.8 − 2.28i)25-s + 17.4i·29-s − 48.3·31-s + (43.6 + 2.00i)35-s + 66.2i·37-s + 14.7i·41-s − 28.4i·43-s − 35.9·47-s + ⋯ |
| L(s) = 1 | + (−0.0458 + 0.998i)5-s − 1.24i·7-s + 0.953i·11-s + 0.414i·13-s − 1.54·17-s + 0.142·19-s + 1.44·23-s + (−0.995 − 0.0915i)25-s + 0.602i·29-s − 1.56·31-s + (1.24 + 0.0572i)35-s + 1.79i·37-s + 0.359i·41-s − 0.661i·43-s − 0.764·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.614i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.789 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8099794302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8099794302\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.229 - 4.99i)T \) |
| good | 7 | \( 1 + 8.73iT - 49T^{2} \) |
| 11 | \( 1 - 10.4iT - 121T^{2} \) |
| 13 | \( 1 - 5.38iT - 169T^{2} \) |
| 17 | \( 1 + 26.2T + 289T^{2} \) |
| 19 | \( 1 - 2.70T + 361T^{2} \) |
| 23 | \( 1 - 33.2T + 529T^{2} \) |
| 29 | \( 1 - 17.4iT - 841T^{2} \) |
| 31 | \( 1 + 48.3T + 961T^{2} \) |
| 37 | \( 1 - 66.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 14.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 28.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 35.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 42.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 55.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 96.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 15.4iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 13.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 63.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 94.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 19.4T + 6.88e3T^{2} \) |
| 89 | \( 1 - 118. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 100. iT - 9.40e3T^{2} \) |
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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
−10.76316992931983002128596702515, −9.788507030390641063362000267875, −8.965003834294115127766050304295, −7.65409311692330677360564191198, −6.97792499476209942165435871772, −6.54593790732215082874910037739, −4.93093544432469243043433512689, −4.09255683325102169790921487172, −3.00299431767854170063669968262, −1.65235532241044813063405729625,
0.27247822111649893783271917147, 1.89967053312408122838757118346, 3.13063080170806990179482072234, 4.47164843934817372484679825373, 5.43809620730264322216964866546, 6.03381436316480370647885244354, 7.32998054463873230889570172114, 8.499594704393535833607690882749, 8.888990430948075740132134356781, 9.540109463482324561324726485358
