| L(s) = 1 | − 5.82i·5-s − 11.4·7-s + 14.6i·11-s − 18.9·13-s + 5.31i·17-s − 8.97·19-s − 16.2i·23-s − 8.97·25-s − 42i·29-s − 9.48·31-s + 66.9i·35-s − 52.9·37-s − 62.9i·41-s + 2.97·43-s + 75.9i·47-s + ⋯ |
| L(s) = 1 | − 1.16i·5-s − 1.64·7-s + 1.33i·11-s − 1.45·13-s + 0.312i·17-s − 0.472·19-s − 0.708i·23-s − 0.358·25-s − 1.44i·29-s − 0.305·31-s + 1.91i·35-s − 1.43·37-s − 1.53i·41-s + 0.0690·43-s + 1.61i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.223335i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.223335i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5.82iT - 25T^{2} \) |
| 7 | \( 1 + 11.4T + 49T^{2} \) |
| 11 | \( 1 - 14.6iT - 121T^{2} \) |
| 13 | \( 1 + 18.9T + 169T^{2} \) |
| 17 | \( 1 - 5.31iT - 289T^{2} \) |
| 19 | \( 1 + 8.97T + 361T^{2} \) |
| 23 | \( 1 + 16.2iT - 529T^{2} \) |
| 29 | \( 1 + 42iT - 841T^{2} \) |
| 31 | \( 1 + 9.48T + 961T^{2} \) |
| 37 | \( 1 + 52.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + 62.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 2.97T + 1.84e3T^{2} \) |
| 47 | \( 1 - 75.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 2.91iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 61.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 119.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 41.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 10.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 20.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 40.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 108. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 106. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 56.8T + 9.40e3T^{2} \) |
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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
−12.09844671071186003940880855036, −10.26772848397734308296110981167, −9.669965175682601209102223205669, −8.854831710587900713715848969275, −7.47795705837422618906243594644, −6.50087109378114592712649509545, −5.14073556379621651848391901673, −4.08912035238932704261451643263, −2.33912834091557295278612245573, −0.11371717290972891094726224560,
2.78713938538179779123173088045, 3.49868970438358132230939231147, 5.47487800578571047940636358809, 6.64025977040393390302795049219, 7.19670981509660302210587104820, 8.759745376845820830000951615831, 9.834644437663523296258997602128, 10.50441844896802329623228220819, 11.55676827582371842404313092995, 12.61157751104327842122051762215
