| L(s) = 1 | − 8·3-s + 4·5-s + 37·9-s + 40·11-s − 36·13-s − 32·15-s + 40·17-s + 72·19-s + 176·23-s − 109·25-s − 80·27-s + 162·29-s + 16·31-s − 320·33-s − 54·37-s + 288·39-s − 472·41-s + 72·43-s + 148·45-s − 144·47-s − 320·51-s + 486·53-s + 160·55-s − 576·57-s + 648·59-s − 684·61-s − 144·65-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 0.357·5-s + 1.37·9-s + 1.09·11-s − 0.768·13-s − 0.550·15-s + 0.570·17-s + 0.869·19-s + 1.59·23-s − 0.871·25-s − 0.570·27-s + 1.03·29-s + 0.0926·31-s − 1.68·33-s − 0.239·37-s + 1.18·39-s − 1.79·41-s + 0.255·43-s + 0.490·45-s − 0.446·47-s − 0.878·51-s + 1.25·53-s + 0.392·55-s − 1.33·57-s + 1.42·59-s − 1.43·61-s − 0.274·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.381940640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.381940640\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 5 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 40 T + p^{3} T^{2} \) |
| 13 | \( 1 + 36 T + p^{3} T^{2} \) |
| 17 | \( 1 - 40 T + p^{3} T^{2} \) |
| 19 | \( 1 - 72 T + p^{3} T^{2} \) |
| 23 | \( 1 - 176 T + p^{3} T^{2} \) |
| 29 | \( 1 - 162 T + p^{3} T^{2} \) |
| 31 | \( 1 - 16 T + p^{3} T^{2} \) |
| 37 | \( 1 + 54 T + p^{3} T^{2} \) |
| 41 | \( 1 + 472 T + p^{3} T^{2} \) |
| 43 | \( 1 - 72 T + p^{3} T^{2} \) |
| 47 | \( 1 + 144 T + p^{3} T^{2} \) |
| 53 | \( 1 - 486 T + p^{3} T^{2} \) |
| 59 | \( 1 - 648 T + p^{3} T^{2} \) |
| 61 | \( 1 + 684 T + p^{3} T^{2} \) |
| 67 | \( 1 + 216 T + p^{3} T^{2} \) |
| 71 | \( 1 - 608 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1008 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1008 T + p^{3} T^{2} \) |
| 83 | \( 1 + 216 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1040 T + p^{3} T^{2} \) |
| 97 | \( 1 + 936 T + p^{3} T^{2} \) |
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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
−9.344728262213915543226305770259, −8.243037671066118208180556079286, −6.97069269878340366114923645411, −6.76193108880016813117194211676, −5.62880739437241334132187239350, −5.19820236765232567874041308223, −4.26875299637242338910438828532, −3.05226471460744021426721415636, −1.52724823015557594951162239626, −0.66598032437793213767173615614,
0.66598032437793213767173615614, 1.52724823015557594951162239626, 3.05226471460744021426721415636, 4.26875299637242338910438828532, 5.19820236765232567874041308223, 5.62880739437241334132187239350, 6.76193108880016813117194211676, 6.97069269878340366114923645411, 8.243037671066118208180556079286, 9.344728262213915543226305770259
