| L(s) = 1 | + 2·5-s + 7-s − 11-s − 6·13-s + 4·17-s − 8·19-s − 6·23-s − 25-s − 2·29-s − 4·31-s + 2·35-s − 37-s − 7·41-s + 2·43-s − 9·47-s − 6·49-s + 3·53-s − 2·55-s + 12·59-s + 4·61-s − 12·65-s − 7·71-s + 7·73-s − 77-s − 3·83-s + 8·85-s + 12·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 0.301·11-s − 1.66·13-s + 0.970·17-s − 1.83·19-s − 1.25·23-s − 1/5·25-s − 0.371·29-s − 0.718·31-s + 0.338·35-s − 0.164·37-s − 1.09·41-s + 0.304·43-s − 1.31·47-s − 6/7·49-s + 0.412·53-s − 0.269·55-s + 1.56·59-s + 0.512·61-s − 1.48·65-s − 0.830·71-s + 0.819·73-s − 0.113·77-s − 0.329·83-s + 0.867·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.392215727702935988482905100051, −7.78790307165276969482845597819, −6.93610596286674052907325458245, −6.09037614955138614512431406118, −5.35667365720479356016286420004, −4.67375669110578511814790030096, −3.62376221698954049890975125760, −2.32752817979750411282892328426, −1.85774356352190049046929099208, 0,
1.85774356352190049046929099208, 2.32752817979750411282892328426, 3.62376221698954049890975125760, 4.67375669110578511814790030096, 5.35667365720479356016286420004, 6.09037614955138614512431406118, 6.93610596286674052907325458245, 7.78790307165276969482845597819, 8.392215727702935988482905100051
