| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 5·11-s − 3·13-s − 4·14-s + 16-s − 17-s − 6·19-s + 5·22-s − 23-s + 3·26-s + 4·28-s − 9·29-s − 5·31-s − 32-s + 34-s − 2·37-s + 6·38-s − 2·41-s + 43-s − 5·44-s + 46-s + 13·47-s + 9·49-s − 3·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 1.50·11-s − 0.832·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s − 1.37·19-s + 1.06·22-s − 0.208·23-s + 0.588·26-s + 0.755·28-s − 1.67·29-s − 0.898·31-s − 0.176·32-s + 0.171·34-s − 0.328·37-s + 0.973·38-s − 0.312·41-s + 0.152·43-s − 0.753·44-s + 0.147·46-s + 1.89·47-s + 9/7·49-s − 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−9.057747157945472879824438218165, −8.355930680838601690753962231329, −7.66961749761248923619224143336, −7.15606919063925609996883365389, −5.75304984600007987704196017855, −5.09648787580173994454132392752, −4.11203523911621610215174756108, −2.50605893476326607268912550054, −1.83322912505559135458679845227, 0,
1.83322912505559135458679845227, 2.50605893476326607268912550054, 4.11203523911621610215174756108, 5.09648787580173994454132392752, 5.75304984600007987704196017855, 7.15606919063925609996883365389, 7.66961749761248923619224143336, 8.355930680838601690753962231329, 9.057747157945472879824438218165
