| L(s) = 1 | − 4·3-s − 2·7-s + 10·9-s + 12·19-s + 8·21-s + 6·25-s − 20·27-s + 8·29-s + 12·37-s + 8·47-s + 6·49-s − 16·53-s − 48·57-s + 16·59-s − 20·63-s − 24·75-s + 35·81-s − 8·83-s − 32·87-s + 32·103-s − 48·111-s + 16·113-s + 34·121-s + 127-s + 131-s − 24·133-s + 137-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 0.755·7-s + 10/3·9-s + 2.75·19-s + 1.74·21-s + 6/5·25-s − 3.84·27-s + 1.48·29-s + 1.97·37-s + 1.16·47-s + 6/7·49-s − 2.19·53-s − 6.35·57-s + 2.08·59-s − 2.51·63-s − 2.77·75-s + 35/9·81-s − 0.878·83-s − 3.43·87-s + 3.15·103-s − 4.55·111-s + 1.50·113-s + 3.09·121-s + 0.0887·127-s + 0.0873·131-s − 2.08·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.123781266\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.123781266\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T - 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 6 T^{2} + 42 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 34 T^{2} + 514 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 12 T^{2} + 102 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 22 T^{2} + 682 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 82 T^{2} + 2722 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 102 T^{2} + 5130 T^{4} - 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 24 T^{2} + 3774 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 132 T^{2} + 10710 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 144 T^{2} + 10830 T^{4} - 144 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 114 T^{2} + 8418 T^{4} - 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 236 T^{2} + 24310 T^{4} - 236 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2 \wr C_2$ | \( 1 - 288 T^{2} + 33150 T^{4} - 288 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 4 T + 102 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 294 T^{2} + 36618 T^{4} - 294 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 204 T^{2} + 28950 T^{4} - 204 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.91011986528721586399771082853, −6.52150724173690645471708573662, −6.31855275639410702315190507325, −6.28896000860352226025167950057, −5.87972296949063765393782322068, −5.74028782784380603773252728103, −5.61389357822136724136234429348, −5.52811665873643998212994601566, −5.00341239518922836017698192603, −4.90390205939182227464557877729, −4.63884101097916424146149123477, −4.51634717324006356244897783507, −4.48995842424915327374043780830, −3.77773525367131207825232594979, −3.75367748425564844978026577241, −3.33036838841329216239080138925, −3.27186977650580084886127669876, −2.81058993810904649015915106080, −2.56375626902082812126707286007, −2.34995531400563220483803751186, −1.65091388823882080071309704809, −1.48349689592514403211363225787, −0.900896108658626997398673603435, −0.67057495725587579823207856294, −0.64854587076054971189441186493,
0.64854587076054971189441186493, 0.67057495725587579823207856294, 0.900896108658626997398673603435, 1.48349689592514403211363225787, 1.65091388823882080071309704809, 2.34995531400563220483803751186, 2.56375626902082812126707286007, 2.81058993810904649015915106080, 3.27186977650580084886127669876, 3.33036838841329216239080138925, 3.75367748425564844978026577241, 3.77773525367131207825232594979, 4.48995842424915327374043780830, 4.51634717324006356244897783507, 4.63884101097916424146149123477, 4.90390205939182227464557877729, 5.00341239518922836017698192603, 5.52811665873643998212994601566, 5.61389357822136724136234429348, 5.74028782784380603773252728103, 5.87972296949063765393782322068, 6.28896000860352226025167950057, 6.31855275639410702315190507325, 6.52150724173690645471708573662, 6.91011986528721586399771082853
