| L(s) = 1 | − 4·5-s + 96·9-s + 96·11-s − 176·19-s + 56·25-s + 280·29-s − 208·31-s − 360·41-s − 384·45-s − 384·55-s + 1.13e3·59-s + 936·61-s − 1.94e3·71-s − 1.92e3·79-s + 5.45e3·81-s − 3.30e3·89-s + 704·95-s + 9.21e3·99-s − 2.32e3·101-s − 680·109-s + 2.78e3·121-s − 884·125-s + 127-s + 131-s + 137-s + 139-s − 1.12e3·145-s + ⋯ |
| L(s) = 1 | − 0.357·5-s + 32/9·9-s + 2.63·11-s − 2.12·19-s + 0.447·25-s + 1.79·29-s − 1.20·31-s − 1.37·41-s − 1.27·45-s − 0.941·55-s + 2.50·59-s + 1.96·61-s − 3.24·71-s − 2.74·79-s + 7.48·81-s − 3.93·89-s + 0.760·95-s + 9.35·99-s − 2.29·101-s − 0.597·109-s + 2.09·121-s − 0.632·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s − 0.641·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.321880653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.321880653\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T - 8 p T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( ( 1 - 16 p T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $D_{4}$ | \( ( 1 - 48 T + 2062 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 2048 T^{2} + 8213778 T^{4} - 2048 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12348 T^{2} + 83683910 T^{4} - 12348 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 88 T + 15360 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 48228 T^{2} + 877537958 T^{4} - 48228 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 140 T + 52502 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 104 T + 61110 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 108716 T^{2} + 6254620278 T^{4} - 108716 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 180 T + 3646 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 135164 T^{2} + 9063402198 T^{4} - 135164 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 354924 T^{2} + 52958993702 T^{4} - 354924 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 138620 T^{2} + 36325958358 T^{4} - 138620 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 568 T + 335888 T^{2} - 568 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 468 T + 353192 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 182636 T^{2} + 164284952598 T^{4} - 182636 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 972 T + 950842 T^{2} + 972 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 574892 T^{2} + 285497185350 T^{4} - 574892 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 964 T + 483402 T^{2} + 964 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1072928 T^{2} + 718495235058 T^{4} - 1072928 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 1652 T + 1922870 T^{2} + 1652 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2626300 T^{2} + 3385412890758 T^{4} - 2626300 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.81629313949541924235334102500, −6.79114679062543326998302014762, −6.56249969902607585621555480188, −6.29328767580400667095758415186, −5.90436097922208638905647589395, −5.57203828135327366451482051876, −5.31512177850556650429459835049, −5.23420258257211694529036374550, −4.68193187209432994059469142540, −4.40925957419844359281048253779, −4.34734934654554268451106318143, −4.16046728782619920318753271938, −4.04957797433837780193371140273, −3.77668485314071591217161611732, −3.76495819747214677203820989284, −3.01804928937911480886375717680, −2.94919466040546737895828760927, −2.55894113674516268954232886831, −1.92248211514876157679609676645, −1.86966899476064065404793268721, −1.59542868772149539834279590582, −1.23856580090479786959067993306, −1.15401492816686136104840107695, −0.806474854653187484704797901552, −0.14630075231618496167787385254,
0.14630075231618496167787385254, 0.806474854653187484704797901552, 1.15401492816686136104840107695, 1.23856580090479786959067993306, 1.59542868772149539834279590582, 1.86966899476064065404793268721, 1.92248211514876157679609676645, 2.55894113674516268954232886831, 2.94919466040546737895828760927, 3.01804928937911480886375717680, 3.76495819747214677203820989284, 3.77668485314071591217161611732, 4.04957797433837780193371140273, 4.16046728782619920318753271938, 4.34734934654554268451106318143, 4.40925957419844359281048253779, 4.68193187209432994059469142540, 5.23420258257211694529036374550, 5.31512177850556650429459835049, 5.57203828135327366451482051876, 5.90436097922208638905647589395, 6.29328767580400667095758415186, 6.56249969902607585621555480188, 6.79114679062543326998302014762, 6.81629313949541924235334102500
