| L(s) = 1 | + 8·7-s − 2·9-s − 8·17-s + 16·23-s + 8·25-s − 24·31-s + 24·41-s − 16·47-s + 16·49-s − 16·63-s + 48·71-s − 16·73-s − 40·79-s + 3·81-s − 8·89-s + 8·97-s + 8·103-s + 24·113-s − 64·119-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 16·153-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 2/3·9-s − 1.94·17-s + 3.33·23-s + 8/5·25-s − 4.31·31-s + 3.74·41-s − 2.33·47-s + 16/7·49-s − 2.01·63-s + 5.69·71-s − 1.87·73-s − 4.50·79-s + 1/3·81-s − 0.847·89-s + 0.812·97-s + 0.788·103-s + 2.25·113-s − 5.86·119-s + 3.27·121-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 + 1.29·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.357940587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.357940587\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 34 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 4354 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 12 T + 96 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1366 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_4$ | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 4066 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 10870 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 24 T + 278 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 20 T + 240 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 12886 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 97 | $D_{4}$ | \( ( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.86219544879729193268236531421, −6.59201299881828062643265550193, −6.26121947452536303248036694556, −5.96431332839633585578637011046, −5.95378892975004486847653585040, −5.53256636586702657274805994317, −5.25776965184202928497629176289, −5.21257076605588841353768201564, −4.96531371720447236051604204426, −4.74820959246706365763185651217, −4.60898812251931788802516369023, −4.52658214244835027700213485213, −4.18755519772350732052040507493, −3.82810363846197742029265266128, −3.42297991747308763712519065650, −3.42021298445737023451488307758, −3.00573779890433975733097045612, −2.66954469873067575026387657305, −2.48840206132413182436529400434, −1.98320913006221134631679663812, −1.97952953842009477607474181692, −1.57772741526845491161086173725, −1.29437511196760295176668582230, −0.930560314517740518581942210336, −0.39582559537767615330989319349,
0.39582559537767615330989319349, 0.930560314517740518581942210336, 1.29437511196760295176668582230, 1.57772741526845491161086173725, 1.97952953842009477607474181692, 1.98320913006221134631679663812, 2.48840206132413182436529400434, 2.66954469873067575026387657305, 3.00573779890433975733097045612, 3.42021298445737023451488307758, 3.42297991747308763712519065650, 3.82810363846197742029265266128, 4.18755519772350732052040507493, 4.52658214244835027700213485213, 4.60898812251931788802516369023, 4.74820959246706365763185651217, 4.96531371720447236051604204426, 5.21257076605588841353768201564, 5.25776965184202928497629176289, 5.53256636586702657274805994317, 5.95378892975004486847653585040, 5.96431332839633585578637011046, 6.26121947452536303248036694556, 6.59201299881828062643265550193, 6.86219544879729193268236531421
