| L(s) = 1 | + 3-s − 1.68i·5-s + 9-s − 3.53i·11-s − 2.93i·13-s − 1.68i·15-s + 2.01i·17-s + 1.69·19-s + 1.59i·23-s + 2.16·25-s + 27-s + 7.94·29-s − 4.95·31-s − 3.53i·33-s − 10.4·37-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.753i·5-s + 0.333·9-s − 1.06i·11-s − 0.812i·13-s − 0.434i·15-s + 0.487i·17-s + 0.389·19-s + 0.333i·23-s + 0.432·25-s + 0.192·27-s + 1.47·29-s − 0.889·31-s − 0.614i·33-s − 1.72·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.986618865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.986618865\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 1.68iT - 5T^{2} \) |
| 11 | \( 1 + 3.53iT - 11T^{2} \) |
| 13 | \( 1 + 2.93iT - 13T^{2} \) |
| 17 | \( 1 - 2.01iT - 17T^{2} \) |
| 19 | \( 1 - 1.69T + 19T^{2} \) |
| 23 | \( 1 - 1.59iT - 23T^{2} \) |
| 29 | \( 1 - 7.94T + 29T^{2} \) |
| 31 | \( 1 + 4.95T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 2.86iT - 41T^{2} \) |
| 43 | \( 1 + 11.7iT - 43T^{2} \) |
| 47 | \( 1 + 6.70T + 47T^{2} \) |
| 53 | \( 1 - 2.92T + 53T^{2} \) |
| 59 | \( 1 + 7.75T + 59T^{2} \) |
| 61 | \( 1 + 12.5iT - 61T^{2} \) |
| 67 | \( 1 + 1.70iT - 67T^{2} \) |
| 71 | \( 1 - 6.13iT - 71T^{2} \) |
| 73 | \( 1 + 2.43iT - 73T^{2} \) |
| 79 | \( 1 + 0.865iT - 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 15.9iT - 89T^{2} \) |
| 97 | \( 1 - 9.82iT - 97T^{2} \) |
| 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
−8.644508623742134418501939969692, −8.274950318197329996889947168814, −7.39106663788110997474345868157, −6.45842119052672861616457319736, −5.48516375953400382198717975763, −4.91047237003195121900182302507, −3.67997388518629920880349338206, −3.12568087775161857595044735065, −1.80426289732509973409288503985, −0.63198979858857738614498090285,
1.51245503948935770875956896603, 2.54832094054360471954389346382, 3.29886500293680413260623788443, 4.39054847971347789501021386092, 5.03658751862523397227150601151, 6.38467686778457265142509165414, 6.93721491636422134491410539813, 7.52909259528206349040579998265, 8.453891230855366440030087771924, 9.231237892227218645647492693754
