| L(s) = 1 | + 4·9-s − 16·23-s + 4·25-s + 32·47-s + 28·49-s + 48·71-s − 16·73-s + 7·81-s − 48·97-s − 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 4/3·9-s − 3.33·23-s + 4/5·25-s + 4.66·47-s + 4·49-s + 5.69·71-s − 1.87·73-s + 7/9·81-s − 4.87·97-s − 2.18·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-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
| \(L(1)\) |
\(\approx\) |
\(4.146244307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.146244307\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 36 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 156 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
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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
−6.85153866718181281366528285608, −6.72175253642393160329086050210, −6.25960204029362143842404245042, −6.07026991624816301116196546047, −5.65942182250206169698231072932, −5.63081379335337284742690797158, −5.61506709198642828891222231239, −5.46296157522766276366388933113, −4.91570752426659730738634242400, −4.70282538881853030453428557524, −4.35217973711021216271720565551, −4.27276073337075573304067774885, −4.04929950229180360976355977422, −3.73147980194962270142867006084, −3.72708384324553953696022622201, −3.68222778967776905529119643772, −2.75435307379947195805181575954, −2.69562431349053212716901617038, −2.51475974876372517779612561254, −2.22546443284183555040368897689, −1.94029420009479756439665917981, −1.61017655821101794869878385782, −1.10838271084235557111784262997, −0.846266450321152904558089093536, −0.45123259966904894137290485982,
0.45123259966904894137290485982, 0.846266450321152904558089093536, 1.10838271084235557111784262997, 1.61017655821101794869878385782, 1.94029420009479756439665917981, 2.22546443284183555040368897689, 2.51475974876372517779612561254, 2.69562431349053212716901617038, 2.75435307379947195805181575954, 3.68222778967776905529119643772, 3.72708384324553953696022622201, 3.73147980194962270142867006084, 4.04929950229180360976355977422, 4.27276073337075573304067774885, 4.35217973711021216271720565551, 4.70282538881853030453428557524, 4.91570752426659730738634242400, 5.46296157522766276366388933113, 5.61506709198642828891222231239, 5.63081379335337284742690797158, 5.65942182250206169698231072932, 6.07026991624816301116196546047, 6.25960204029362143842404245042, 6.72175253642393160329086050210, 6.85153866718181281366528285608
