| L(s) = 1 | − 4·3-s + 2·5-s + 10·9-s − 8·15-s + 10·17-s − 16·19-s − 25-s − 20·27-s + 12·31-s + 20·45-s + 21·49-s − 40·51-s + 64·57-s + 16·59-s − 10·61-s − 8·67-s − 16·71-s + 18·73-s + 4·75-s − 4·79-s + 35·81-s + 20·85-s − 48·93-s − 32·95-s + 36·101-s − 28·103-s + 8·107-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 0.894·5-s + 10/3·9-s − 2.06·15-s + 2.42·17-s − 3.67·19-s − 1/5·25-s − 3.84·27-s + 2.15·31-s + 2.98·45-s + 3·49-s − 5.60·51-s + 8.47·57-s + 2.08·59-s − 1.28·61-s − 0.977·67-s − 1.89·71-s + 2.10·73-s + 0.461·75-s − 0.450·79-s + 35/9·81-s + 2.16·85-s − 4.97·93-s − 3.28·95-s + 3.58·101-s − 2.75·103-s + 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.895370797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.895370797\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| good | 5 | $D_{4}$ | \( ( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3 p T^{2} + 200 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 576 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 350 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 5 T + 32 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T^{2} - 66 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 426 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 6206 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 3648 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 169 T^{2} + 11484 T^{4} - 169 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 6006 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 + 5 T + 120 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 - 9 T + 92 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 2 T + 126 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.06192087274228902801985120585, −5.86863286485460344604228139262, −5.63496159089670454858487777701, −5.53138354181609709283848671363, −5.44190516491237406829524029159, −5.13716559796510178774713217236, −4.80014771447471940370307214340, −4.60777882992524230117807493876, −4.58584482033290902410095045860, −4.24376562135573241620136209342, −4.11401816176568410977028216705, −3.96764876536958267130292578021, −3.79487663212530819106619672532, −3.31192593420858052323894264772, −3.00108542774397664678244841986, −2.87368551646971438685952505441, −2.76740086842857847388025583367, −2.05777015518477270953930011399, −1.92152701358001627327760603063, −1.92060744259310102309015794400, −1.78948282612848675309691070043, −1.02553839776104416761535986845, −0.865714006245449104706532923834, −0.64346358949394989925940538632, −0.41669240776022289053565060875,
0.41669240776022289053565060875, 0.64346358949394989925940538632, 0.865714006245449104706532923834, 1.02553839776104416761535986845, 1.78948282612848675309691070043, 1.92060744259310102309015794400, 1.92152701358001627327760603063, 2.05777015518477270953930011399, 2.76740086842857847388025583367, 2.87368551646971438685952505441, 3.00108542774397664678244841986, 3.31192593420858052323894264772, 3.79487663212530819106619672532, 3.96764876536958267130292578021, 4.11401816176568410977028216705, 4.24376562135573241620136209342, 4.58584482033290902410095045860, 4.60777882992524230117807493876, 4.80014771447471940370307214340, 5.13716559796510178774713217236, 5.44190516491237406829524029159, 5.53138354181609709283848671363, 5.63496159089670454858487777701, 5.86863286485460344604228139262, 6.06192087274228902801985120585
