| L(s) = 1 | − 2i·2-s + i·3-s − 2·4-s + 2·6-s − 2i·7-s + 2·9-s + 11-s − 2i·12-s − 4i·13-s − 4·14-s − 4·16-s − 2i·17-s − 4i·18-s + 2·21-s − 2i·22-s + i·23-s + ⋯ |
| L(s) = 1 | − 1.41i·2-s + 0.577i·3-s − 4-s + 0.816·6-s − 0.755i·7-s + 0.666·9-s + 0.301·11-s − 0.577i·12-s − 1.10i·13-s − 1.06·14-s − 16-s − 0.485i·17-s − 0.942i·18-s + 0.436·21-s − 0.426i·22-s + 0.208i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.685976 - 1.10993i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.685976 - 1.10993i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 2iT - 2T^{2} \) |
| 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 3iT - 37T^{2} \) |
| 41 | \( 1 + 8T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 5T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 + 7iT - 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
| 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
−11.43558853428207847962679445284, −10.53213938107971233805540401914, −10.06665428588082261076458198889, −9.251477131161992413121569130863, −7.84237457796866250230965741845, −6.65108822145973853100510609900, −4.90979256355504269162896996230, −3.96908582963553082050621433685, −2.94748151665832611682961435334, −1.14477533681318858572605649219,
2.05327083745803521896391993931, 4.23102062652037313826257902463, 5.45129199703095345621440390589, 6.55003528032179780820142489133, 7.01651101672238661047656276596, 8.209335702871744089617690896496, 8.889224296787787090698632950554, 10.03750022045939640447522428704, 11.53396745702261126366409928745, 12.25940612211621853665452299995
