| L(s) = 1 | + 2·2-s + 4·4-s + 2·7-s + 8·8-s − 70·11-s − 54·13-s + 4·14-s + 16·16-s − 22·17-s + 24·19-s − 140·22-s − 100·23-s − 108·26-s + 8·28-s − 216·29-s + 208·31-s + 32·32-s − 44·34-s + 254·37-s + 48·38-s + 206·41-s − 292·43-s − 280·44-s − 200·46-s − 320·47-s − 339·49-s − 216·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.107·7-s + 0.353·8-s − 1.91·11-s − 1.15·13-s + 0.0763·14-s + 1/4·16-s − 0.313·17-s + 0.289·19-s − 1.35·22-s − 0.906·23-s − 0.814·26-s + 0.0539·28-s − 1.38·29-s + 1.20·31-s + 0.176·32-s − 0.221·34-s + 1.12·37-s + 0.204·38-s + 0.784·41-s − 1.03·43-s − 0.959·44-s − 0.641·46-s − 0.993·47-s − 0.988·49-s − 0.576·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 70 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 22 T + p^{3} T^{2} \) |
| 19 | \( 1 - 24 T + p^{3} T^{2} \) |
| 23 | \( 1 + 100 T + p^{3} T^{2} \) |
| 29 | \( 1 + 216 T + p^{3} T^{2} \) |
| 31 | \( 1 - 208 T + p^{3} T^{2} \) |
| 37 | \( 1 - 254 T + p^{3} T^{2} \) |
| 41 | \( 1 - 206 T + p^{3} T^{2} \) |
| 43 | \( 1 + 292 T + p^{3} T^{2} \) |
| 47 | \( 1 + 320 T + p^{3} T^{2} \) |
| 53 | \( 1 + 402 T + p^{3} T^{2} \) |
| 59 | \( 1 - 370 T + p^{3} T^{2} \) |
| 61 | \( 1 + 550 T + p^{3} T^{2} \) |
| 67 | \( 1 + 728 T + p^{3} T^{2} \) |
| 71 | \( 1 - 540 T + p^{3} T^{2} \) |
| 73 | \( 1 + 604 T + p^{3} T^{2} \) |
| 79 | \( 1 - 792 T + p^{3} T^{2} \) |
| 83 | \( 1 - 404 T + p^{3} T^{2} \) |
| 89 | \( 1 - 938 T + p^{3} T^{2} \) |
| 97 | \( 1 + 56 T + p^{3} T^{2} \) |
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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
−10.30738529281219340529384958067, −9.565781803646040750201508407263, −8.002620034804854150153640753093, −7.58895092690028014822206377155, −6.29073839751339634421066641982, −5.25469291660954935325258950477, −4.55914480399969525270586930644, −3.06334673250803879995167419470, −2.12825028786074897279987242645, 0,
2.12825028786074897279987242645, 3.06334673250803879995167419470, 4.55914480399969525270586930644, 5.25469291660954935325258950477, 6.29073839751339634421066641982, 7.58895092690028014822206377155, 8.002620034804854150153640753093, 9.565781803646040750201508407263, 10.30738529281219340529384958067
