L(s) = 1 | − 0.680·7-s + 1.68·11-s − 5.14·13-s − 1.31·17-s − 0.324·19-s + 3.78·23-s + 8.64·29-s − 4.14·31-s − 1.35·37-s + 7.15·41-s + 7.29·43-s − 12.9·47-s − 6.53·49-s − 8.83·53-s − 8.81·59-s + 9.97·61-s + 4.17·67-s + 0.891·71-s + 7.82·73-s − 1.14·77-s + 9.65·79-s + 4.85·83-s − 17.4·89-s + 3.50·91-s − 8.93·97-s + 10.8·101-s − 11.6·103-s + ⋯ |
L(s) = 1 | − 0.257·7-s + 0.506·11-s − 1.42·13-s − 0.320·17-s − 0.0744·19-s + 0.790·23-s + 1.60·29-s − 0.744·31-s − 0.222·37-s + 1.11·41-s + 1.11·43-s − 1.89·47-s − 0.933·49-s − 1.21·53-s − 1.14·59-s + 1.27·61-s + 0.510·67-s + 0.105·71-s + 0.915·73-s − 0.130·77-s + 1.08·79-s + 0.533·83-s − 1.85·89-s + 0.366·91-s − 0.907·97-s + 1.08·101-s − 1.15·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 0.680T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 + 5.14T + 13T^{2} \) |
| 17 | \( 1 + 1.31T + 17T^{2} \) |
| 19 | \( 1 + 0.324T + 19T^{2} \) |
| 23 | \( 1 - 3.78T + 23T^{2} \) |
| 29 | \( 1 - 8.64T + 29T^{2} \) |
| 31 | \( 1 + 4.14T + 31T^{2} \) |
| 37 | \( 1 + 1.35T + 37T^{2} \) |
| 41 | \( 1 - 7.15T + 41T^{2} \) |
| 43 | \( 1 - 7.29T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 8.83T + 53T^{2} \) |
| 59 | \( 1 + 8.81T + 59T^{2} \) |
| 61 | \( 1 - 9.97T + 61T^{2} \) |
| 67 | \( 1 - 4.17T + 67T^{2} \) |
| 71 | \( 1 - 0.891T + 71T^{2} \) |
| 73 | \( 1 - 7.82T + 73T^{2} \) |
| 79 | \( 1 - 9.65T + 79T^{2} \) |
| 83 | \( 1 - 4.85T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 + 8.93T + 97T^{2} \) |
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
−7.48369500254062751897195672676, −6.68218101811703885060454974484, −6.33758484107047460082019552177, −5.19370950570824984585935498538, −4.78318570469792800809332639777, −3.92403998176683022503862785656, −3.00168405481693189380235743242, −2.34447782051954629856768543659, −1.23146979048232082764371199717, 0,
1.23146979048232082764371199717, 2.34447782051954629856768543659, 3.00168405481693189380235743242, 3.92403998176683022503862785656, 4.78318570469792800809332639777, 5.19370950570824984585935498538, 6.33758484107047460082019552177, 6.68218101811703885060454974484, 7.48369500254062751897195672676