| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 10-s + 2·11-s − 6·13-s − 16-s − 6·17-s + 6·19-s − 20-s − 2·22-s + 4·23-s + 25-s + 6·26-s − 8·29-s + 6·31-s − 5·32-s + 6·34-s − 6·37-s − 6·38-s + 3·40-s − 6·41-s − 2·44-s − 4·46-s − 50-s + 6·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s + 0.603·11-s − 1.66·13-s − 1/4·16-s − 1.45·17-s + 1.37·19-s − 0.223·20-s − 0.426·22-s + 0.834·23-s + 1/5·25-s + 1.17·26-s − 1.48·29-s + 1.07·31-s − 0.883·32-s + 1.02·34-s − 0.986·37-s − 0.973·38-s + 0.474·40-s − 0.937·41-s − 0.301·44-s − 0.589·46-s − 0.141·50-s + 0.832·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.929131819633424032481983386190, −7.952525594805363297976652459498, −7.21129932781375417432404118744, −6.58583184575665717452070168487, −5.21594225200877212023113542647, −4.86075960403071897586907134866, −3.75278266753238330801144288814, −2.50383922446396710571631521160, −1.42009345245993264838225109650, 0,
1.42009345245993264838225109650, 2.50383922446396710571631521160, 3.75278266753238330801144288814, 4.86075960403071897586907134866, 5.21594225200877212023113542647, 6.58583184575665717452070168487, 7.21129932781375417432404118744, 7.952525594805363297976652459498, 8.929131819633424032481983386190
