| L(s) = 1 | + 3-s + 5-s + 9-s − 2·13-s + 15-s + 6·17-s − 4·19-s − 8·23-s + 25-s + 27-s − 2·29-s − 8·31-s − 2·37-s − 2·39-s + 6·41-s + 45-s + 6·51-s + 6·53-s − 4·57-s − 4·59-s + 6·61-s − 2·65-s + 8·67-s − 8·69-s + 14·73-s + 75-s − 8·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 1.45·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.328·37-s − 0.320·39-s + 0.937·41-s + 0.149·45-s + 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.768·61-s − 0.248·65-s + 0.977·67-s − 0.963·69-s + 1.63·73-s + 0.115·75-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−15.80786863111672, −14.90425193625605, −14.65128396886378, −14.15848869621854, −13.72310134556802, −12.93703252068893, −12.55524439373957, −12.10112155988428, −11.36702617647010, −10.64595120694073, −10.16124757090304, −9.647511074655057, −9.210298224396440, −8.435322152338282, −7.952777356618569, −7.406724045229381, −6.787586093504978, −5.957308504657399, −5.555761710944029, −4.813065936440907, −3.897046637252034, −3.591840683237225, −2.517825791888902, −2.091114471844770, −1.230479211160076, 0,
1.230479211160076, 2.091114471844770, 2.517825791888902, 3.591840683237225, 3.897046637252034, 4.813065936440907, 5.555761710944029, 5.957308504657399, 6.787586093504978, 7.406724045229381, 7.952777356618569, 8.435322152338282, 9.210298224396440, 9.647511074655057, 10.16124757090304, 10.64595120694073, 11.36702617647010, 12.10112155988428, 12.55524439373957, 12.93703252068893, 13.72310134556802, 14.15848869621854, 14.65128396886378, 14.90425193625605, 15.80786863111672
