| L(s) = 1 | + 3-s − 2·5-s + 9-s − 11-s − 2·13-s − 2·15-s − 17-s − 4·19-s − 4·23-s − 25-s + 27-s + 6·29-s + 8·31-s − 33-s − 2·37-s − 2·39-s + 6·41-s + 4·43-s − 2·45-s − 4·47-s − 51-s + 2·53-s + 2·55-s − 4·57-s + 4·59-s − 10·61-s + 4·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.516·15-s − 0.242·17-s − 0.917·19-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s − 0.174·33-s − 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.609·43-s − 0.298·45-s − 0.583·47-s − 0.140·51-s + 0.274·53-s + 0.269·55-s − 0.529·57-s + 0.520·59-s − 1.28·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 219912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 219912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.664920351\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664920351\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.94183480726151, −12.39698821610341, −12.16227757486519, −11.63684455659126, −11.15117526057179, −10.54447178627378, −10.22290420857809, −9.687302206263791, −9.205540542693775, −8.582746046662281, −8.142344924698656, −7.944467852372006, −7.375569057484695, −6.815802121131535, −6.323892841274827, −5.855736515572210, −4.941005864188228, −4.666729525421891, −4.061763586441183, −3.709751557082633, −2.964642446032231, −2.444586564566619, −2.031910543432882, −1.071166884239452, −0.3752496529681827,
0.3752496529681827, 1.071166884239452, 2.031910543432882, 2.444586564566619, 2.964642446032231, 3.709751557082633, 4.061763586441183, 4.666729525421891, 4.941005864188228, 5.855736515572210, 6.323892841274827, 6.815802121131535, 7.375569057484695, 7.944467852372006, 8.142344924698656, 8.582746046662281, 9.205540542693775, 9.687302206263791, 10.22290420857809, 10.54447178627378, 11.15117526057179, 11.63684455659126, 12.16227757486519, 12.39698821610341, 12.94183480726151
