| L(s) = 1 | + 1.73i·3-s + 2.23·5-s − 11.2i·7-s − 2.99·9-s − 11.2i·11-s + 17.4·13-s + 3.87i·15-s + 1.41·17-s − 1.63i·19-s + 19.4·21-s + 22.4i·23-s + 5.00·25-s − 5.19i·27-s + 52.2·29-s − 29.3i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.447·5-s − 1.60i·7-s − 0.333·9-s − 1.01i·11-s + 1.33·13-s + 0.258i·15-s + 0.0833·17-s − 0.0860i·19-s + 0.924·21-s + 0.974i·23-s + 0.200·25-s − 0.192i·27-s + 1.80·29-s − 0.946i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.866 + 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.63492 - 0.438076i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63492 - 0.438076i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 - 2.23T \) |
| good | 7 | \( 1 + 11.2iT - 49T^{2} \) |
| 11 | \( 1 + 11.2iT - 121T^{2} \) |
| 13 | \( 1 - 17.4T + 169T^{2} \) |
| 17 | \( 1 - 1.41T + 289T^{2} \) |
| 19 | \( 1 + 1.63iT - 361T^{2} \) |
| 23 | \( 1 - 22.4iT - 529T^{2} \) |
| 29 | \( 1 - 52.2T + 841T^{2} \) |
| 31 | \( 1 + 29.3iT - 961T^{2} \) |
| 37 | \( 1 + 53.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 42T + 1.68e3T^{2} \) |
| 43 | \( 1 + 55.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 44.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 88.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 30.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 59.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 18.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 115. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 58.4T + 5.32e3T^{2} \) |
| 79 | \( 1 - 15.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 36.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 92.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 139.T + 9.40e3T^{2} \) |
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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
−11.48078180729168245231562410288, −10.75754874376975260450044845998, −10.10900255130328324815234852231, −8.956773821044750659735550822406, −7.969336414026586962552689036434, −6.68910099611592003177088881025, −5.65791367220523057174850050735, −4.23154026843251376941037221213, −3.31241622642405416943313720306, −1.02139749668131859751379209424,
1.69385275369153948251624200135, 2.90853070081733176967898625348, 4.85036281478121097017111113668, 5.99635146315209069905548062317, 6.73794108173106434683075529132, 8.291429822114396122699599468532, 8.862724847930813366015340379284, 10.00232425329967982714008275147, 11.15234390248090092127463737516, 12.31985970111388522141401304531
