| L(s) = 1 | + 3-s + (−1 − 2i)5-s + 2i·7-s + 9-s − 6i·11-s − 2·13-s + (−1 − 2i)15-s + 6i·17-s + 4i·19-s + 2i·21-s + 8i·23-s + (−3 + 4i)25-s + 27-s − 8·31-s − 6i·33-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (−0.447 − 0.894i)5-s + 0.755i·7-s + 0.333·9-s − 1.80i·11-s − 0.554·13-s + (−0.258 − 0.516i)15-s + 1.45i·17-s + 0.917i·19-s + 0.436i·21-s + 1.66i·23-s + (−0.600 + 0.800i)25-s + 0.192·27-s − 1.43·31-s − 1.04i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.437587836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.437587836\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (1 + 2i)T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
| 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
−8.829584486892682325279613103536, −7.84370602085598289035840888683, −7.65942778005623696973776287030, −6.09324126502811829392224599480, −5.77664470785569002585189088426, −4.91651765509695897031941335120, −3.68002430109632250045186902585, −3.48562117117840399441614015921, −2.12324509069957569588633335662, −1.16754944604682028026514655090,
0.40164717097590757484998935695, 2.13026047433678039284921860868, 2.65760611288065635864675629291, 3.69398762171163444952696398128, 4.52377631543394273727610367039, 4.99559030762216562900428125268, 6.56079091901228065161528580486, 7.07956395228963760949152284517, 7.39868286171354696447874837833, 8.134515067791440563334750344702
