| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s − 11-s − 15-s + 4·17-s + 4·19-s − 2·21-s + 4·23-s + 25-s + 27-s + 6·29-s − 33-s + 2·35-s + 6·37-s + 10·41-s + 2·43-s − 45-s − 4·47-s − 3·49-s + 4·51-s − 2·53-s + 55-s + 4·57-s + 12·59-s − 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 0.258·15-s + 0.970·17-s + 0.917·19-s − 0.436·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.174·33-s + 0.338·35-s + 0.986·37-s + 1.56·41-s + 0.304·43-s − 0.149·45-s − 0.583·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s + 0.134·55-s + 0.529·57-s + 1.56·59-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.797805057\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.797805057\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−9.641626984537129846362591387392, −8.854514197506430374257929320197, −7.936470180953667064067257559734, −7.36355796853793679007010598732, −6.43485493806038581563599137006, −5.42623329727972043184980842327, −4.38649357117939208592741270275, −3.33042731439276962650014222981, −2.71534072624835604330487811219, −0.989027028478981179665033157492,
0.989027028478981179665033157492, 2.71534072624835604330487811219, 3.33042731439276962650014222981, 4.38649357117939208592741270275, 5.42623329727972043184980842327, 6.43485493806038581563599137006, 7.36355796853793679007010598732, 7.936470180953667064067257559734, 8.854514197506430374257929320197, 9.641626984537129846362591387392
