L(s) = 1 | + 2i·2-s − 4·4-s + 4i·7-s − 8i·8-s − 42·11-s + 20i·13-s − 8·14-s + 16·16-s + 93i·17-s − 59·19-s − 84i·22-s − 9i·23-s − 40·26-s − 16i·28-s + 120·29-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.215i·7-s − 0.353i·8-s − 1.15·11-s + 0.426i·13-s − 0.152·14-s + 0.250·16-s + 1.32i·17-s − 0.712·19-s − 0.814i·22-s − 0.0815i·23-s − 0.301·26-s − 0.107i·28-s + 0.768·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4442435886\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4442435886\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4iT - 343T^{2} \) |
| 11 | \( 1 + 42T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 93iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 59T + 6.85e3T^{2} \) |
| 23 | \( 1 + 9iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 120T + 2.43e4T^{2} \) |
| 31 | \( 1 - 47T + 2.97e4T^{2} \) |
| 37 | \( 1 - 262iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 126T + 6.89e4T^{2} \) |
| 43 | \( 1 + 178iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 144iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 741iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 444T + 2.05e5T^{2} \) |
| 61 | \( 1 - 221T + 2.26e5T^{2} \) |
| 67 | \( 1 - 538iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 690T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.12e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 665T + 4.93e5T^{2} \) |
| 83 | \( 1 + 75iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.08e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.54e3iT - 9.12e5T^{2} \) |
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
−8.695330745545629638251939905814, −8.398255430826609363871896383805, −7.48572047568283395908595317996, −6.56375464954475299796980265158, −5.88389194161566846052451513438, −4.95795002596260180582402137739, −4.14299164207318261494357643866, −2.94738287272598796437081323314, −1.73120459552926602631661226311, −0.11999934690286510692210981930,
0.921303647761441561430164235795, 2.37318411345268648256282068789, 3.01816786585987169132873685716, 4.24011120762946700024291930720, 5.04097745584322394053695444204, 5.87689611565618829051033469911, 7.09848037487535711891494507473, 7.83231379331928827472721699745, 8.673464280337089200725686505514, 9.496926176292796861430375662078