| L(s) = 1 | + 4·3-s − 6·5-s − 7·7-s − 11·9-s − 12·11-s + 82·13-s − 24·15-s − 30·17-s + 68·19-s − 28·21-s − 216·23-s − 89·25-s − 152·27-s − 246·29-s + 112·31-s − 48·33-s + 42·35-s − 110·37-s + 328·39-s − 246·41-s − 172·43-s + 66·45-s − 192·47-s + 49·49-s − 120·51-s − 558·53-s + 72·55-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 0.536·5-s − 0.377·7-s − 0.407·9-s − 0.328·11-s + 1.74·13-s − 0.413·15-s − 0.428·17-s + 0.821·19-s − 0.290·21-s − 1.95·23-s − 0.711·25-s − 1.08·27-s − 1.57·29-s + 0.648·31-s − 0.253·33-s + 0.202·35-s − 0.488·37-s + 1.34·39-s − 0.937·41-s − 0.609·43-s + 0.218·45-s − 0.595·47-s + 1/7·49-s − 0.329·51-s − 1.44·53-s + 0.176·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 82 T + p^{3} T^{2} \) |
| 17 | \( 1 + 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 68 T + p^{3} T^{2} \) |
| 23 | \( 1 + 216 T + p^{3} T^{2} \) |
| 29 | \( 1 + 246 T + p^{3} T^{2} \) |
| 31 | \( 1 - 112 T + p^{3} T^{2} \) |
| 37 | \( 1 + 110 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 192 T + p^{3} T^{2} \) |
| 53 | \( 1 + 558 T + p^{3} T^{2} \) |
| 59 | \( 1 - 540 T + p^{3} T^{2} \) |
| 61 | \( 1 + 110 T + p^{3} T^{2} \) |
| 67 | \( 1 - 140 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 + 550 T + p^{3} T^{2} \) |
| 79 | \( 1 - 208 T + p^{3} T^{2} \) |
| 83 | \( 1 - 516 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1398 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1586 T + p^{3} T^{2} \) |
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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
−10.12528604205771189110666401735, −9.215823791984288944729899750293, −8.284646168257653410387205714789, −7.82442186692745137546595442365, −6.45479564422681832295287720107, −5.55314249092004707563323961682, −3.91205579060244468303258770839, −3.33920704810777841973751625046, −1.88179001471311342318053272741, 0,
1.88179001471311342318053272741, 3.33920704810777841973751625046, 3.91205579060244468303258770839, 5.55314249092004707563323961682, 6.45479564422681832295287720107, 7.82442186692745137546595442365, 8.284646168257653410387205714789, 9.215823791984288944729899750293, 10.12528604205771189110666401735
