| L(s) = 1 | − 0.631·2-s − 3-s − 1.60·4-s + 1.65·5-s + 0.631·6-s + 2.27·8-s + 9-s − 1.04·10-s + 2.82·11-s + 1.60·12-s + 3.15·13-s − 1.65·15-s + 1.76·16-s − 1.11·17-s − 0.631·18-s − 3.11·19-s − 2.65·20-s − 1.78·22-s − 6.51·23-s − 2.27·24-s − 2.24·25-s − 1.99·26-s − 27-s + 9.16·29-s + 1.04·30-s + 4.35·31-s − 5.66·32-s + ⋯ |
| L(s) = 1 | − 0.446·2-s − 0.577·3-s − 0.800·4-s + 0.742·5-s + 0.257·6-s + 0.803·8-s + 0.333·9-s − 0.331·10-s + 0.850·11-s + 0.462·12-s + 0.876·13-s − 0.428·15-s + 0.442·16-s − 0.271·17-s − 0.148·18-s − 0.713·19-s − 0.594·20-s − 0.379·22-s − 1.35·23-s − 0.463·24-s − 0.449·25-s − 0.391·26-s − 0.192·27-s + 1.70·29-s + 0.191·30-s + 0.781·31-s − 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.631T + 2T^{2} \) |
| 5 | \( 1 - 1.65T + 5T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 3.15T + 13T^{2} \) |
| 17 | \( 1 + 1.11T + 17T^{2} \) |
| 19 | \( 1 + 3.11T + 19T^{2} \) |
| 23 | \( 1 + 6.51T + 23T^{2} \) |
| 29 | \( 1 - 9.16T + 29T^{2} \) |
| 31 | \( 1 - 4.35T + 31T^{2} \) |
| 37 | \( 1 + 11.1T + 37T^{2} \) |
| 43 | \( 1 + 0.684T + 43T^{2} \) |
| 47 | \( 1 + 6.17T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 + 7.71T + 59T^{2} \) |
| 61 | \( 1 + 0.419T + 61T^{2} \) |
| 67 | \( 1 - 9.95T + 67T^{2} \) |
| 71 | \( 1 - 7.39T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 0.195T + 79T^{2} \) |
| 83 | \( 1 - 2.50T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
| 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
−8.068638550804700693784319840119, −6.75425350016614257096987870692, −6.36742105728774781439561713330, −5.67623338137104768551554235445, −4.78304354596603205951594088652, −4.20742322931222392056672554198, −3.37432997746338060557494224213, −1.92293739266139938144572286158, −1.25451133973638491739122497524, 0,
1.25451133973638491739122497524, 1.92293739266139938144572286158, 3.37432997746338060557494224213, 4.20742322931222392056672554198, 4.78304354596603205951594088652, 5.67623338137104768551554235445, 6.36742105728774781439561713330, 6.75425350016614257096987870692, 8.068638550804700693784319840119
