L(s) = 1 | + 2.32i·2-s + i·3-s − 3.42·4-s + (1.29 − 1.82i)5-s − 2.32·6-s + i·7-s − 3.32i·8-s − 9-s + (4.24 + 3.01i)10-s + 11-s − 3.42i·12-s + 0.502i·13-s − 2.32·14-s + (1.82 + 1.29i)15-s + 0.894·16-s + 4.71i·17-s + ⋯ |
L(s) = 1 | + 1.64i·2-s + 0.577i·3-s − 1.71·4-s + (0.578 − 0.815i)5-s − 0.951·6-s + 0.377i·7-s − 1.17i·8-s − 0.333·9-s + (1.34 + 0.952i)10-s + 0.301·11-s − 0.989i·12-s + 0.139i·13-s − 0.622·14-s + (0.470 + 0.333i)15-s + 0.223·16-s + 1.14i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 + 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.093345224\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.093345224\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-1.29 + 1.82i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 2.32iT - 2T^{2} \) |
| 13 | \( 1 - 0.502iT - 13T^{2} \) |
| 17 | \( 1 - 4.71iT - 17T^{2} \) |
| 19 | \( 1 + 5.19T + 19T^{2} \) |
| 23 | \( 1 - 8.45iT - 23T^{2} \) |
| 29 | \( 1 + 6.83T + 29T^{2} \) |
| 31 | \( 1 + 3.74T + 31T^{2} \) |
| 37 | \( 1 - 10.6iT - 37T^{2} \) |
| 41 | \( 1 - 4.41T + 41T^{2} \) |
| 43 | \( 1 - 2.08iT - 43T^{2} \) |
| 47 | \( 1 + 3.12iT - 47T^{2} \) |
| 53 | \( 1 + 6.43iT - 53T^{2} \) |
| 59 | \( 1 + 9.33T + 59T^{2} \) |
| 61 | \( 1 - 14.8T + 61T^{2} \) |
| 67 | \( 1 - 8.31iT - 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 - 1.67iT - 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 - 5.93iT - 83T^{2} \) |
| 89 | \( 1 - 4.63T + 89T^{2} \) |
| 97 | \( 1 - 6.94iT - 97T^{2} \) |
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
−9.822949651328418769891339903681, −9.280117639582616557553092105807, −8.556803361205621378125422867117, −8.015320797785913824474716694138, −6.87388657057167049863893010540, −5.95362025694958283422587044236, −5.53972059848012466909163580451, −4.61863086102033991022128809591, −3.76648549065850720095059687282, −1.84168029919866533072943538582,
0.45670322277372144816685848527, 1.94619564143001454754687952948, 2.55947723045974012267633879700, 3.59680786187324650560889872205, 4.57034730834741093730645867264, 5.85830079191330379569402600458, 6.79412268544304061463587638465, 7.58143338716190976829891373777, 8.915353523759408531659134089691, 9.387601649037217415811551947481