L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 7·5-s − 6·6-s + 7·7-s − 8·8-s + 9·9-s + 14·10-s + 11·11-s + 12·12-s − 67·13-s − 14·14-s − 21·15-s + 16·16-s + 30·17-s − 18·18-s − 7·19-s − 28·20-s + 21·21-s − 22·22-s + 28·23-s − 24·24-s − 76·25-s + 134·26-s + 27·27-s + 28·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.626·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.442·10-s + 0.301·11-s + 0.288·12-s − 1.42·13-s − 0.267·14-s − 0.361·15-s + 1/4·16-s + 0.428·17-s − 0.235·18-s − 0.0845·19-s − 0.313·20-s + 0.218·21-s − 0.213·22-s + 0.253·23-s − 0.204·24-s − 0.607·25-s + 1.01·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
good | 5 | \( 1 + 7 T + p^{3} T^{2} \) |
| 13 | \( 1 + 67 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 + 7 T + p^{3} T^{2} \) |
| 23 | \( 1 - 28 T + p^{3} T^{2} \) |
| 29 | \( 1 - 121 T + p^{3} T^{2} \) |
| 31 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 71 T + p^{3} T^{2} \) |
| 41 | \( 1 + 180 T + p^{3} T^{2} \) |
| 43 | \( 1 + 108 T + p^{3} T^{2} \) |
| 47 | \( 1 - 71 T + p^{3} T^{2} \) |
| 53 | \( 1 - 128 T + p^{3} T^{2} \) |
| 59 | \( 1 + 429 T + p^{3} T^{2} \) |
| 61 | \( 1 - 22 T + p^{3} T^{2} \) |
| 67 | \( 1 + 803 T + p^{3} T^{2} \) |
| 71 | \( 1 - 468 T + p^{3} T^{2} \) |
| 73 | \( 1 + 117 T + p^{3} T^{2} \) |
| 79 | \( 1 + 96 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1122 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1146 T + p^{3} T^{2} \) |
| 97 | \( 1 + 92 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.02093661223488596513011609690, −9.260594361098412231083034496616, −8.339840832471571152812023329613, −7.57066452820735548638508763045, −6.92161295198282181215241191329, −5.39096213503668459424458951989, −4.16601178026925040226042863566, −2.93770681074307696211309829245, −1.67182386761785152109276720802, 0,
1.67182386761785152109276720802, 2.93770681074307696211309829245, 4.16601178026925040226042863566, 5.39096213503668459424458951989, 6.92161295198282181215241191329, 7.57066452820735548638508763045, 8.339840832471571152812023329613, 9.260594361098412231083034496616, 10.02093661223488596513011609690