| L(s) = 1 | − 8·4-s − 36·11-s + 48·16-s − 86·19-s + 60·29-s + 296·31-s − 60·41-s + 288·44-s + 347·49-s − 48·59-s + 1.22e3·61-s − 256·64-s + 312·71-s + 688·76-s + 1.03e3·79-s − 3.07e3·89-s − 4.62e3·101-s − 1.24e3·109-s − 480·116-s − 752·121-s − 2.36e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s − 0.986·11-s + 3/4·16-s − 1.03·19-s + 0.384·29-s + 1.71·31-s − 0.228·41-s + 0.986·44-s + 1.01·49-s − 0.105·59-s + 2.57·61-s − 1/2·64-s + 0.521·71-s + 1.03·76-s + 1.47·79-s − 3.65·89-s − 4.55·101-s − 1.09·109-s − 0.384·116-s − 0.564·121-s − 1.71·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2550754929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2550754929\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 347 T^{2} + 185928 T^{4} - 347 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 18 T + 862 T^{2} + 18 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 7823 T^{2} + 24930408 T^{4} - 7823 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 15872 T^{2} + 111187518 T^{4} - 15872 p^{6} T^{6} + p^{12} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 43 T + 13710 T^{2} + 43 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24908 T^{2} + 320075718 T^{4} - 24908 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 30 T + 47122 T^{2} - 30 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 148 T + 63177 T^{2} - 148 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 40730 T^{2} + 1964009643 T^{4} - 40730 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 30 T + 121138 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 133298 T^{2} + 17012554803 T^{4} - 133298 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 270680 T^{2} + 35119990158 T^{4} - 270680 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 53312 T^{2} - 20927563410 T^{4} - 53312 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 24 T + 380806 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 613 T + 205092 T^{2} - 613 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 236219 T^{2} + 35052520272 T^{4} - 236219 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 156 T + 112462 T^{2} - 156 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 454586 T^{2} + 223496476827 T^{4} - 454586 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 259 T + p^{3} T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 + 1330312 T^{2} + 1088439166878 T^{4} + 1330312 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 1536 T + 1728898 T^{2} + 1536 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2577503 T^{2} + 3040417246080 T^{4} - 2577503 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.57654338228858704560598937700, −6.20881737627533355873758788103, −6.03025726953473494940088453302, −5.66403152846029951089865148945, −5.53026864995980754795442872510, −5.36303249304395732935007747803, −5.29234816712380338926199255023, −4.70912497795344952883389320725, −4.65321723112800303564406273621, −4.58357659425470976487060491594, −4.32379009106807816318677831636, −3.77400953950211517213130134867, −3.75683648923666058505165291151, −3.70864558615097883075193906288, −3.33204156347800335667785914015, −2.62288000588241945924426012474, −2.58113428504070179206041344890, −2.53894803813424052661727547206, −2.47444034339802652334367476259, −1.71082022027357647676080327069, −1.42672484593955734795086531160, −1.14214959887977250782530211598, −0.919719581809418831292575252085, −0.41362054448941331859228846517, −0.085509201564606671783903166246,
0.085509201564606671783903166246, 0.41362054448941331859228846517, 0.919719581809418831292575252085, 1.14214959887977250782530211598, 1.42672484593955734795086531160, 1.71082022027357647676080327069, 2.47444034339802652334367476259, 2.53894803813424052661727547206, 2.58113428504070179206041344890, 2.62288000588241945924426012474, 3.33204156347800335667785914015, 3.70864558615097883075193906288, 3.75683648923666058505165291151, 3.77400953950211517213130134867, 4.32379009106807816318677831636, 4.58357659425470976487060491594, 4.65321723112800303564406273621, 4.70912497795344952883389320725, 5.29234816712380338926199255023, 5.36303249304395732935007747803, 5.53026864995980754795442872510, 5.66403152846029951089865148945, 6.03025726953473494940088453302, 6.20881737627533355873758788103, 6.57654338228858704560598937700
