| L(s) = 1 | + 4·5-s + 8·9-s + 8·11-s + 4·19-s + 2·25-s + 8·29-s + 16·31-s − 8·41-s + 32·45-s + 4·49-s + 32·55-s + 16·59-s + 30·81-s + 8·89-s + 16·95-s + 64·99-s − 32·101-s + 8·109-s + 12·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 8/3·9-s + 2.41·11-s + 0.917·19-s + 2/5·25-s + 1.48·29-s + 2.87·31-s − 1.24·41-s + 4.77·45-s + 4/7·49-s + 4.31·55-s + 2.08·59-s + 10/3·81-s + 0.847·89-s + 1.64·95-s + 6.43·99-s − 3.18·101-s + 0.766·109-s + 1.09·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.776330416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.776330416\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 706 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 646 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 136 T^{2} + 7330 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 4310 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11014 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 4066 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 36 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 25030 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2134 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 216 T^{2} + 23282 T^{4} - 216 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.54459091350667474916127375640, −6.91371484383468952760536954479, −6.76287551397612015035854901870, −6.75244988571815910970089269782, −6.58409211853966635224205939266, −6.34855433802452763196846781008, −6.22159150786997629898258269138, −5.91250364726114413921427842003, −5.52155135059482610383661610814, −5.15343044506616206376739297898, −5.11442390211032529899716056014, −4.85532898234289611865066330598, −4.55124669646651139632100676316, −4.11317253808898792369126266616, −4.00813364884349270312630731997, −3.80299108773671577181222091368, −3.76337759880744972274493669689, −3.05674410845415407159707716459, −2.67260273746830861350575717014, −2.52588013236837656241344902724, −2.15046904564781886536363772481, −1.55759012049151117337035451335, −1.38487909258119473222844343303, −1.24243521776005975096593263135, −0.943973784696653802894924334232,
0.943973784696653802894924334232, 1.24243521776005975096593263135, 1.38487909258119473222844343303, 1.55759012049151117337035451335, 2.15046904564781886536363772481, 2.52588013236837656241344902724, 2.67260273746830861350575717014, 3.05674410845415407159707716459, 3.76337759880744972274493669689, 3.80299108773671577181222091368, 4.00813364884349270312630731997, 4.11317253808898792369126266616, 4.55124669646651139632100676316, 4.85532898234289611865066330598, 5.11442390211032529899716056014, 5.15343044506616206376739297898, 5.52155135059482610383661610814, 5.91250364726114413921427842003, 6.22159150786997629898258269138, 6.34855433802452763196846781008, 6.58409211853966635224205939266, 6.75244988571815910970089269782, 6.76287551397612015035854901870, 6.91371484383468952760536954479, 7.54459091350667474916127375640
