| L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s − 11-s − 6·13-s + 2·15-s − 4·19-s + 4·21-s + 4·23-s − 25-s − 27-s − 6·29-s + 33-s + 8·35-s − 6·37-s + 6·39-s + 6·41-s − 4·43-s − 2·45-s + 12·47-s + 9·49-s + 2·53-s + 2·55-s + 4·57-s − 12·59-s + 14·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.516·15-s − 0.917·19-s + 0.872·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.174·33-s + 1.35·35-s − 0.986·37-s + 0.960·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s + 1.75·47-s + 9/7·49-s + 0.274·53-s + 0.269·55-s + 0.529·57-s − 1.56·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.65275592928295, −13.08189002455909, −12.77755904607432, −12.40322110761250, −11.96783062866174, −11.52965523065638, −10.91442786344520, −10.43288394104337, −10.04011773590305, −9.572549188145291, −8.979251517514515, −8.666229134250772, −7.624554482763494, −7.481237721678367, −7.057578701246696, −6.461224702788608, −5.952248051022991, −5.407870145760980, −4.801851021924241, −4.305928539382685, −3.689043018648928, −3.245323110086154, −2.512671029415290, −2.054766978503949, −0.8772163001667894, 0, 0,
0.8772163001667894, 2.054766978503949, 2.512671029415290, 3.245323110086154, 3.689043018648928, 4.305928539382685, 4.801851021924241, 5.407870145760980, 5.952248051022991, 6.461224702788608, 7.057578701246696, 7.481237721678367, 7.624554482763494, 8.666229134250772, 8.979251517514515, 9.572549188145291, 10.04011773590305, 10.43288394104337, 10.91442786344520, 11.52965523065638, 11.96783062866174, 12.40322110761250, 12.77755904607432, 13.08189002455909, 13.65275592928295
