| L(s) = 1 | − 7-s − 3·9-s − 11-s − 6·13-s + 4·17-s − 4·19-s − 4·23-s + 2·31-s − 2·37-s + 10·41-s − 8·43-s + 49-s − 2·53-s − 14·61-s + 3·63-s + 2·67-s + 4·71-s − 4·73-s + 77-s − 14·79-s + 9·81-s − 18·83-s + 14·89-s + 6·91-s + 2·97-s + 3·99-s + 101-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 9-s − 0.301·11-s − 1.66·13-s + 0.970·17-s − 0.917·19-s − 0.834·23-s + 0.359·31-s − 0.328·37-s + 1.56·41-s − 1.21·43-s + 1/7·49-s − 0.274·53-s − 1.79·61-s + 0.377·63-s + 0.244·67-s + 0.474·71-s − 0.468·73-s + 0.113·77-s − 1.57·79-s + 81-s − 1.97·83-s + 1.48·89-s + 0.628·91-s + 0.203·97-s + 0.301·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.96333766043794, −13.25433082054202, −12.64942449219664, −12.42450036409914, −11.84885389264037, −11.51939447369767, −10.81600637297106, −10.30261713319879, −9.938456698091837, −9.467510886634573, −8.893312169457941, −8.345678353351978, −7.851503395742410, −7.433435342497644, −6.858687757480037, −6.159372911020489, −5.805473092078773, −5.270599067004052, −4.615110116841700, −4.212962215594160, −3.234479326401618, −2.984255876919543, −2.292434818926810, −1.756968532401174, −0.6075880817924796, 0,
0.6075880817924796, 1.756968532401174, 2.292434818926810, 2.984255876919543, 3.234479326401618, 4.212962215594160, 4.615110116841700, 5.270599067004052, 5.805473092078773, 6.159372911020489, 6.858687757480037, 7.433435342497644, 7.851503395742410, 8.345678353351978, 8.893312169457941, 9.467510886634573, 9.938456698091837, 10.30261713319879, 10.81600637297106, 11.51939447369767, 11.84885389264037, 12.42450036409914, 12.64942449219664, 13.25433082054202, 13.96333766043794
