| L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s − 7-s + 8-s + 9-s + 2·10-s + 11-s + 12-s − 2·13-s − 14-s + 2·15-s + 16-s − 2·17-s + 18-s + 2·20-s − 21-s + 22-s + 24-s − 25-s − 2·26-s + 27-s − 28-s − 2·29-s + 2·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.447·20-s − 0.218·21-s + 0.213·22-s + 0.204·24-s − 1/5·25-s − 0.392·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.757859414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.757859414\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−11.07234697320449720415471226055, −10.02727748724745217892398596458, −9.432611216215457888567824586170, −8.354764684441443853355726283697, −7.16036475066804637810786253712, −6.34557563328331303195978146208, −5.33211397644666962563070735994, −4.18774926881798203669654007783, −2.96523175330119569799799482797, −1.88425382150531611231140922836,
1.88425382150531611231140922836, 2.96523175330119569799799482797, 4.18774926881798203669654007783, 5.33211397644666962563070735994, 6.34557563328331303195978146208, 7.16036475066804637810786253712, 8.354764684441443853355726283697, 9.432611216215457888567824586170, 10.02727748724745217892398596458, 11.07234697320449720415471226055
