L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 1.73·7-s + 8-s + 9-s + 10-s − 12-s − 6.46·13-s + 1.73·14-s − 15-s + 16-s − 4.26·17-s + 18-s − 6.46·19-s + 20-s − 1.73·21-s + 8.19·23-s − 24-s + 25-s − 6.46·26-s − 27-s + 1.73·28-s − 6.46·29-s − 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.654·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.288·12-s − 1.79·13-s + 0.462·14-s − 0.258·15-s + 0.250·16-s − 1.03·17-s + 0.235·18-s − 1.48·19-s + 0.223·20-s − 0.377·21-s + 1.70·23-s − 0.204·24-s + 0.200·25-s − 1.26·26-s − 0.192·27-s + 0.327·28-s − 1.20·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 7 | \( 1 - 1.73T + 7T^{2} \) |
| 13 | \( 1 + 6.46T + 13T^{2} \) |
| 17 | \( 1 + 4.26T + 17T^{2} \) |
| 19 | \( 1 + 6.46T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 + 6.46T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 7.19T + 37T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 43 | \( 1 + 4.73T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 2.19T + 53T^{2} \) |
| 59 | \( 1 - 2.19T + 59T^{2} \) |
| 61 | \( 1 + 1.26T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 7.39T + 71T^{2} \) |
| 73 | \( 1 + 1.26T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 5.53T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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
−8.014684757049097371758214995810, −7.05199973841545401532310316882, −6.80756788566478105013425310855, −5.73345137640759434287799783386, −4.98216074358931462677950766580, −4.70176258550413658560010032588, −3.60182253210008652045459675120, −2.35091975179907651328230673856, −1.78779427008832037712290472200, 0,
1.78779427008832037712290472200, 2.35091975179907651328230673856, 3.60182253210008652045459675120, 4.70176258550413658560010032588, 4.98216074358931462677950766580, 5.73345137640759434287799783386, 6.80756788566478105013425310855, 7.05199973841545401532310316882, 8.014684757049097371758214995810