| L(s) = 1 | + 4·5-s − 6·9-s + 6·11-s + 10·19-s + 5·25-s + 16·29-s − 4·31-s + 12·41-s − 24·45-s − 2·49-s + 24·55-s + 8·59-s − 12·61-s + 24·71-s − 28·79-s + 9·81-s + 4·89-s + 40·95-s − 36·99-s + 4·109-s + 31·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2·9-s + 1.80·11-s + 2.29·19-s + 25-s + 2.97·29-s − 0.718·31-s + 1.87·41-s − 3.57·45-s − 2/7·49-s + 3.23·55-s + 1.04·59-s − 1.53·61-s + 2.84·71-s − 3.15·79-s + 81-s + 0.423·89-s + 4.10·95-s − 3.61·99-s + 0.383·109-s + 2.81·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.727164127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.727164127\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 3 T^{2} - 520 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 45 T^{2} - 184 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.962686256396124794997185488102, −7.32627038122318878141957151154, −7.25400471037142305563566122526, −7.13306027940299045540808387390, −6.72531254960470363944281516232, −6.34950570998274108940591651543, −6.23791852655595223827498015136, −6.17563815887789357044969176875, −5.77046415430889980623262016103, −5.72574568996022039296566000170, −5.28459725291016964542831959648, −5.21967989225295330136914365667, −4.95366397005094649059406455067, −4.52362580491294833034020751662, −4.12106222277381466252963932499, −4.08641743454540912391561741134, −3.51424921635709369385715730583, −3.15130592821623598128012225223, −2.95499721526651732964333005001, −2.79300864418129792760058174939, −2.44087868296226965242062633353, −1.77292869520538079212190014310, −1.76541162097098055124785547300, −0.909388870489807595718687289013, −0.889577415937023522694604967108,
0.889577415937023522694604967108, 0.909388870489807595718687289013, 1.76541162097098055124785547300, 1.77292869520538079212190014310, 2.44087868296226965242062633353, 2.79300864418129792760058174939, 2.95499721526651732964333005001, 3.15130592821623598128012225223, 3.51424921635709369385715730583, 4.08641743454540912391561741134, 4.12106222277381466252963932499, 4.52362580491294833034020751662, 4.95366397005094649059406455067, 5.21967989225295330136914365667, 5.28459725291016964542831959648, 5.72574568996022039296566000170, 5.77046415430889980623262016103, 6.17563815887789357044969176875, 6.23791852655595223827498015136, 6.34950570998274108940591651543, 6.72531254960470363944281516232, 7.13306027940299045540808387390, 7.25400471037142305563566122526, 7.32627038122318878141957151154, 7.962686256396124794997185488102
