| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 6·13-s − 15-s − 2·17-s − 4·19-s − 4·21-s − 8·23-s + 25-s − 27-s + 6·29-s + 4·35-s + 6·37-s − 6·39-s + 10·41-s + 4·43-s + 45-s + 8·47-s + 9·49-s + 2·51-s − 10·53-s + 4·57-s − 6·61-s + 4·63-s + 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.66·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s − 0.872·21-s − 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.676·35-s + 0.986·37-s − 0.960·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s + 0.280·51-s − 1.37·53-s + 0.529·57-s − 0.768·61-s + 0.503·63-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.760568811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.760568811\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.29585778371206426685174091423, −9.082691646086291194533627292052, −8.338450200049268291048738447687, −7.64493796343622540106320647343, −6.25442273107448540439252397130, −5.92809906508794084871962623137, −4.67467612086321652791764027956, −4.05477376880357436694530570270, −2.25919867640796924053383456153, −1.20192952395638838673519576448,
1.20192952395638838673519576448, 2.25919867640796924053383456153, 4.05477376880357436694530570270, 4.67467612086321652791764027956, 5.92809906508794084871962623137, 6.25442273107448540439252397130, 7.64493796343622540106320647343, 8.338450200049268291048738447687, 9.082691646086291194533627292052, 10.29585778371206426685174091423
