| L(s) = 1 | + 4·3-s + 8·9-s − 12·11-s − 8·19-s + 4·27-s − 48·33-s + 12·41-s − 32·57-s − 4·73-s − 30·81-s + 20·97-s − 96·99-s + 12·107-s − 36·113-s + 72·121-s + 48·123-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s − 64·171-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s − 3.61·11-s − 1.83·19-s + 0.769·27-s − 8.35·33-s + 1.87·41-s − 4.23·57-s − 0.468·73-s − 3.33·81-s + 2.03·97-s − 9.64·99-s + 1.16·107-s − 3.38·113-s + 6.54·121-s + 4.32·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s − 4.89·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.368173064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.368173064\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 2 T^{2} + p^{2} T^{4} ) \) |
| 5 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 + 14 T^{2} + p^{2} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 46 T^{2} + p^{2} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 142 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 - 94 T^{2} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.24315209609916002423608578396, −6.86873610371920548949351954611, −6.81682269050190471935727306525, −6.34369895410471039823898820054, −6.27251631120177624135918902903, −5.86678970338225293798722999571, −5.60990862814417219174587597023, −5.53166792335997803092802365183, −5.20976492418083289269618805835, −5.05689201055559865323964542205, −4.71113053908061087341608248141, −4.33068361284429287816300816593, −4.16639541035850498437946511141, −4.09125170335544069538260575300, −3.82059982014977416317612823008, −3.16313210337679842745745687320, −3.02327752875602756576090677175, −2.95816015912365599026498917975, −2.83398950151287235668794246703, −2.28432107220857578681260564428, −2.16327308110333458672797745456, −1.95380210996197667645067908734, −1.74049096653222721103113602336, −0.834254082288875586561506245995, −0.29573520450173864128791400019,
0.29573520450173864128791400019, 0.834254082288875586561506245995, 1.74049096653222721103113602336, 1.95380210996197667645067908734, 2.16327308110333458672797745456, 2.28432107220857578681260564428, 2.83398950151287235668794246703, 2.95816015912365599026498917975, 3.02327752875602756576090677175, 3.16313210337679842745745687320, 3.82059982014977416317612823008, 4.09125170335544069538260575300, 4.16639541035850498437946511141, 4.33068361284429287816300816593, 4.71113053908061087341608248141, 5.05689201055559865323964542205, 5.20976492418083289269618805835, 5.53166792335997803092802365183, 5.60990862814417219174587597023, 5.86678970338225293798722999571, 6.27251631120177624135918902903, 6.34369895410471039823898820054, 6.81682269050190471935727306525, 6.86873610371920548949351954611, 7.24315209609916002423608578396
