| L(s) = 1 | − 3-s − 4·5-s − 2·7-s + 9-s + 11-s + 4·15-s − 6·17-s − 4·19-s + 2·21-s − 6·23-s + 11·25-s − 27-s − 6·29-s − 33-s + 8·35-s − 6·37-s − 10·41-s + 8·43-s − 4·45-s + 6·47-s − 3·49-s + 6·51-s + 12·53-s − 4·55-s + 4·57-s + 8·59-s − 4·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.78·5-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 1.03·15-s − 1.45·17-s − 0.917·19-s + 0.436·21-s − 1.25·23-s + 11/5·25-s − 0.192·27-s − 1.11·29-s − 0.174·33-s + 1.35·35-s − 0.986·37-s − 1.56·41-s + 1.21·43-s − 0.596·45-s + 0.875·47-s − 3/7·49-s + 0.840·51-s + 1.64·53-s − 0.539·55-s + 0.529·57-s + 1.04·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3567718876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3567718876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.876300872321245146558644250499, −8.400023062643409926759897781983, −7.38183121231392422955020128238, −6.85160516446561167635362014810, −6.10555524724608198736821229927, −4.94271345259324786254649457064, −3.98854111110819735222504713742, −3.70588708733816310650814183486, −2.21912342023084737394252260741, −0.38250580765381455986530330042,
0.38250580765381455986530330042, 2.21912342023084737394252260741, 3.70588708733816310650814183486, 3.98854111110819735222504713742, 4.94271345259324786254649457064, 6.10555524724608198736821229927, 6.85160516446561167635362014810, 7.38183121231392422955020128238, 8.400023062643409926759897781983, 8.876300872321245146558644250499
