| L(s) = 1 | − i·3-s − 3.46i·5-s + 7-s − 9-s + 5.46i·11-s − 2i·13-s − 3.46·15-s + 7.46·17-s + 6.92i·19-s − i·21-s − 5.46·23-s − 6.99·25-s + i·27-s − 8.92i·29-s + 2.92·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.54i·5-s + 0.377·7-s − 0.333·9-s + 1.64i·11-s − 0.554i·13-s − 0.894·15-s + 1.81·17-s + 1.58i·19-s − 0.218i·21-s − 1.13·23-s − 1.39·25-s + 0.192i·27-s − 1.65i·29-s + 0.525·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.709235969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.709235969\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 11 | \( 1 - 5.46iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 - 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 5.46T + 23T^{2} \) |
| 29 | \( 1 + 8.92iT - 29T^{2} \) |
| 31 | \( 1 - 2.92T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 4.53T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 2.92T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 14.9iT - 59T^{2} \) |
| 61 | \( 1 + 4.92iT - 61T^{2} \) |
| 67 | \( 1 + 10.9iT - 67T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 - 0.928T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 - 4.92T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.049529663015836596798181728441, −7.54121793407339945642660611308, −6.38497683216916893888872649802, −5.59435029451244762726711532548, −5.12103723144253047213808126394, −4.29005085879892040976498223803, −3.53711592703475040868945207512, −2.01710352525635126858220231923, −1.59583053493359406601502906509, −0.48098954920790350812239599206,
1.16070893419317198944099399089, 2.65102661408536581996405329029, 3.14051916258183323447918206498, 3.76446421702001566996899511801, 4.82090247136823781326549190145, 5.67444671677751706845012387358, 6.24604238738743034252595965741, 7.00156849915268103893531365993, 7.69772673693921853967399350830, 8.464300163057408642349615321158
