L(s) = 1 | − 2.24·2-s + 3.03·4-s + 1.67·5-s + 7-s − 2.31·8-s − 3.74·10-s + 0.0139·11-s − 1.81·13-s − 2.24·14-s − 0.869·16-s + 6.81·17-s + 7.14·19-s + 5.06·20-s − 0.0312·22-s + 0.0905·23-s − 2.20·25-s + 4.06·26-s + 3.03·28-s − 7.70·29-s − 8.56·31-s + 6.58·32-s − 15.2·34-s + 1.67·35-s + 1.24·37-s − 16.0·38-s − 3.87·40-s + 6.73·41-s + ⋯ |
L(s) = 1 | − 1.58·2-s + 1.51·4-s + 0.747·5-s + 0.377·7-s − 0.818·8-s − 1.18·10-s + 0.00419·11-s − 0.503·13-s − 0.599·14-s − 0.217·16-s + 1.65·17-s + 1.63·19-s + 1.13·20-s − 0.00665·22-s + 0.0188·23-s − 0.441·25-s + 0.798·26-s + 0.573·28-s − 1.42·29-s − 1.53·31-s + 1.16·32-s − 2.62·34-s + 0.282·35-s + 0.204·37-s − 2.60·38-s − 0.611·40-s + 1.05·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 5 | \( 1 - 1.67T + 5T^{2} \) |
| 11 | \( 1 - 0.0139T + 11T^{2} \) |
| 13 | \( 1 + 1.81T + 13T^{2} \) |
| 17 | \( 1 - 6.81T + 17T^{2} \) |
| 19 | \( 1 - 7.14T + 19T^{2} \) |
| 23 | \( 1 - 0.0905T + 23T^{2} \) |
| 29 | \( 1 + 7.70T + 29T^{2} \) |
| 31 | \( 1 + 8.56T + 31T^{2} \) |
| 37 | \( 1 - 1.24T + 37T^{2} \) |
| 41 | \( 1 - 6.73T + 41T^{2} \) |
| 43 | \( 1 + 6.87T + 43T^{2} \) |
| 47 | \( 1 + 7.32T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 + 9.03T + 59T^{2} \) |
| 61 | \( 1 + 1.91T + 61T^{2} \) |
| 67 | \( 1 + 2.93T + 67T^{2} \) |
| 71 | \( 1 + 5.54T + 71T^{2} \) |
| 73 | \( 1 + 3.15T + 73T^{2} \) |
| 79 | \( 1 + 0.921T + 79T^{2} \) |
| 83 | \( 1 - 4.65T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 + 8.25T + 97T^{2} \) |
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
−7.68329016166057972783003005301, −7.23811404038264431523504497148, −6.20960780597394742329570875479, −5.53048641872543318840466806535, −4.94742022252762225946177755032, −3.63688214735881721784280486472, −2.81949613533479793744580554461, −1.71362146251955903212444086090, −1.34801238512757822375336792981, 0,
1.34801238512757822375336792981, 1.71362146251955903212444086090, 2.81949613533479793744580554461, 3.63688214735881721784280486472, 4.94742022252762225946177755032, 5.53048641872543318840466806535, 6.20960780597394742329570875479, 7.23811404038264431523504497148, 7.68329016166057972783003005301