| L(s) = 1 | − 4·5-s − 96·11-s + 176·19-s + 56·25-s − 280·29-s + 208·31-s − 360·41-s − 98·49-s + 384·55-s + 1.13e3·59-s − 936·61-s + 1.94e3·71-s − 1.92e3·79-s − 3.30e3·89-s − 704·95-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 + 149-s + 151-s − 832·155-s + ⋯ |
| L(s) = 1 | − 0.357·5-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 − 2/7·49-s + 0.941·55-s + 2.50·59-s − 1.96·61-s + 3.24·71-s − 2.74·79-s − 3.93·89-s − 0.760·95-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 + 0.000549·149-s + 0.000538·151-s − 0.431·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.02251755250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02251755250\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + 4 T - 8 p T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 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
−6.60357044786594700566271684091, −6.27958924125961135583378819702, −6.08659780308148549718065712005, −5.56168462811386510090538029672, −5.55892196244029980861727029960, −5.46578967640920417495488551922, −5.37260931009846798663669034037, −4.96574430858866185204388961497, −4.77549749031232578044846195554, −4.60993526761282217637829802170, −4.31259029213756238516746111430, −3.95795521736868786588147753231, −3.59621954586283896651441333910, −3.45483469015478330098290349019, −3.40870781205256989002022583434, −2.86780495393704588786676916609, −2.57241910187142316521865476139, −2.54796051323242177618993907422, −2.43019899286941032532091952201, −1.84994128538031236005343706884, −1.39021936574120155441955513992, −1.17290056939743858193682682596, −1.06775453937472594466604343850, −0.31906880850278730609167614038, −0.02964197501155196524420197363,
0.02964197501155196524420197363, 0.31906880850278730609167614038, 1.06775453937472594466604343850, 1.17290056939743858193682682596, 1.39021936574120155441955513992, 1.84994128538031236005343706884, 2.43019899286941032532091952201, 2.54796051323242177618993907422, 2.57241910187142316521865476139, 2.86780495393704588786676916609, 3.40870781205256989002022583434, 3.45483469015478330098290349019, 3.59621954586283896651441333910, 3.95795521736868786588147753231, 4.31259029213756238516746111430, 4.60993526761282217637829802170, 4.77549749031232578044846195554, 4.96574430858866185204388961497, 5.37260931009846798663669034037, 5.46578967640920417495488551922, 5.55892196244029980861727029960, 5.56168462811386510090538029672, 6.08659780308148549718065712005, 6.27958924125961135583378819702, 6.60357044786594700566271684091
