| L(s) = 1 | + 3-s + 2·5-s + 7-s + 9-s + 6·11-s + 2·15-s − 17-s + 2·19-s + 21-s − 25-s + 27-s − 4·29-s + 6·33-s + 2·35-s + 8·37-s + 2·41-s + 4·43-s + 2·45-s + 8·47-s + 49-s − 51-s − 14·53-s + 12·55-s + 2·57-s + 6·59-s − 10·61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.80·11-s + 0.516·15-s − 0.242·17-s + 0.458·19-s + 0.218·21-s − 1/5·25-s + 0.192·27-s − 0.742·29-s + 1.04·33-s + 0.338·35-s + 1.31·37-s + 0.312·41-s + 0.609·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.140·51-s − 1.92·53-s + 1.61·55-s + 0.264·57-s + 0.781·59-s − 1.28·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.738063311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.738063311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.134164837338788413639208438017, −7.46603341641157271320583884912, −6.62046066577234178764365192027, −6.09794694248165775418964096315, −5.29493588340823502451977697192, −4.29114994329411998855775531647, −3.76610235290619253102304426024, −2.68782933410282242455774467045, −1.83426129324841325945667964244, −1.10616086819766023945312158628,
1.10616086819766023945312158628, 1.83426129324841325945667964244, 2.68782933410282242455774467045, 3.76610235290619253102304426024, 4.29114994329411998855775531647, 5.29493588340823502451977697192, 6.09794694248165775418964096315, 6.62046066577234178764365192027, 7.46603341641157271320583884912, 8.134164837338788413639208438017
