| L(s) = 1 | − 8·7-s − 3·9-s − 28·17-s − 8·23-s − 10·25-s − 16·31-s − 6·41-s + 8·47-s + 30·49-s + 24·63-s + 32·71-s − 12·73-s − 16·79-s + 24·89-s − 2·97-s + 8·103-s − 4·113-s + 224·119-s − 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 84·153-s + 157-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 9-s − 6.79·17-s − 1.66·23-s − 2·25-s − 2.87·31-s − 0.937·41-s + 1.16·47-s + 30/7·49-s + 3.02·63-s + 3.79·71-s − 1.40·73-s − 1.80·79-s + 2.54·89-s − 0.203·97-s + 0.788·103-s − 0.376·113-s + 20.5·119-s − 1.90·121-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 + 6.79·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 5 T + p T^{2} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 10 T + 71 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 + 37 T^{2} - 2112 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - 90 T^{2} + 1211 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + T - 96 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.36190361563683479198312413726, −7.10825857892951713775742737015, −6.75392567472553002682312124933, −6.73559535660070100796950963029, −6.63800177647304519237626200782, −6.28551888208523037909502203752, −6.24573219751533747367215823476, −6.04993745392372623895262013727, −5.87032629960682260605032151483, −5.55833982843767662372845153475, −5.13423419999117884573344469879, −4.93618791677228579690970693228, −4.86371209358299049337210215971, −4.22785720497696196667211052070, −4.03456380393944183725638612500, −3.98949330804812137268362526078, −3.89881410918223008829561556889, −3.43817703627823871619380424630, −3.39659158085874796893480546075, −2.72277130523598161042220954684, −2.52626708933821913045964312331, −2.24266118522025334928963041377, −2.15956956772968935763049049066, −2.10676770694954226864533411573, −1.37024506425538815992285015538, 0, 0, 0, 0,
1.37024506425538815992285015538, 2.10676770694954226864533411573, 2.15956956772968935763049049066, 2.24266118522025334928963041377, 2.52626708933821913045964312331, 2.72277130523598161042220954684, 3.39659158085874796893480546075, 3.43817703627823871619380424630, 3.89881410918223008829561556889, 3.98949330804812137268362526078, 4.03456380393944183725638612500, 4.22785720497696196667211052070, 4.86371209358299049337210215971, 4.93618791677228579690970693228, 5.13423419999117884573344469879, 5.55833982843767662372845153475, 5.87032629960682260605032151483, 6.04993745392372623895262013727, 6.24573219751533747367215823476, 6.28551888208523037909502203752, 6.63800177647304519237626200782, 6.73559535660070100796950963029, 6.75392567472553002682312124933, 7.10825857892951713775742737015, 7.36190361563683479198312413726
