L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 7-s − 8-s + 9-s − 11-s − 2·12-s − 6.74·13-s + 14-s + 16-s + 6.74·17-s − 18-s − 6.74·19-s + 2·21-s + 22-s + 6.74·23-s + 2·24-s + 6.74·26-s + 4·27-s − 28-s + 8.74·29-s + 4.74·31-s − 32-s + 2·33-s − 6.74·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 0.5·4-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.301·11-s − 0.577·12-s − 1.87·13-s + 0.267·14-s + 0.250·16-s + 1.63·17-s − 0.235·18-s − 1.54·19-s + 0.436·21-s + 0.213·22-s + 1.40·23-s + 0.408·24-s + 1.32·26-s + 0.769·27-s − 0.188·28-s + 1.62·29-s + 0.852·31-s − 0.176·32-s + 0.348·33-s − 1.15·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 2T + 3T^{2} \) |
| 13 | \( 1 + 6.74T + 13T^{2} \) |
| 17 | \( 1 - 6.74T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 - 6.74T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 + 0.744T + 37T^{2} \) |
| 41 | \( 1 + 4T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 4.74T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 1.25T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 6.74T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
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
−8.130014966227299819415813346295, −7.26983555950026925281863795615, −6.69627114659404053778147701216, −6.01987031901699954339518435262, −5.10939266257305154357538387914, −4.68174321122328254398782262910, −3.17165809557258956893501301044, −2.43806178000908702264492520940, −0.997038494175514044118552061593, 0,
0.997038494175514044118552061593, 2.43806178000908702264492520940, 3.17165809557258956893501301044, 4.68174321122328254398782262910, 5.10939266257305154357538387914, 6.01987031901699954339518435262, 6.69627114659404053778147701216, 7.26983555950026925281863795615, 8.130014966227299819415813346295