| L(s) = 1 | + 6·3-s + 5-s + 3·7-s + 15·9-s − 3·11-s + 6·15-s + 9·17-s + 19-s + 18·21-s + 12·23-s + 5·25-s + 18·27-s − 2·29-s − 31-s − 18·33-s + 3·35-s + 27·37-s − 30·41-s + 15·45-s + 15·47-s + 7·49-s + 54·51-s − 3·53-s − 3·55-s + 6·57-s − 59-s + 12·61-s + ⋯ |
| L(s) = 1 | + 3.46·3-s + 0.447·5-s + 1.13·7-s + 5·9-s − 0.904·11-s + 1.54·15-s + 2.18·17-s + 0.229·19-s + 3.92·21-s + 2.50·23-s + 25-s + 3.46·27-s − 0.371·29-s − 0.179·31-s − 3.13·33-s + 0.507·35-s + 4.43·37-s − 4.68·41-s + 2.23·45-s + 2.18·47-s + 49-s + 7.56·51-s − 0.412·53-s − 0.404·55-s + 0.794·57-s − 0.130·59-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.63262179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.63262179\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T^{2} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 + 3 T - T^{2} - 36 T^{3} - 120 T^{4} - 36 p T^{5} - p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 126 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 9 T + 63 T^{2} - 324 T^{3} + 1466 T^{4} - 324 p T^{5} + 63 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - T - 23 T^{2} + 14 T^{3} + 196 T^{4} + 14 p T^{5} - 23 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $D_{4}$ | \( ( 1 + T + 44 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + T - 47 T^{2} - 14 T^{3} + 1312 T^{4} - 14 p T^{5} - 47 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 27 T + 373 T^{2} - 3510 T^{3} + 24522 T^{4} - 3510 p T^{5} + 373 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 15 T + 124 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 125 T^{2} + 7248 T^{4} - 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 15 T + 183 T^{2} - 1620 T^{3} + 12980 T^{4} - 1620 p T^{5} + 183 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 3 T + 67 T^{2} + 192 T^{3} + 1446 T^{4} + 192 p T^{5} + 67 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + T - 103 T^{2} - 14 T^{3} + 7276 T^{4} - 14 p T^{5} - 103 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 12 T + 43 T^{2} + 252 T^{3} - 2304 T^{4} + 252 p T^{5} + 43 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 18 T + 193 T^{2} - 1530 T^{3} + 9972 T^{4} - 1530 p T^{5} + 193 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 94 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 15 T + 235 T^{2} + 2400 T^{3} + 25746 T^{4} + 2400 p T^{5} + 235 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 7 T - 107 T^{2} - 14 T^{3} + 14224 T^{4} - 14 p T^{5} - 107 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 245 T^{2} + 28656 T^{4} - 245 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 7 T - 40 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 146 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−8.022779928948390959193001048335, −7.66455122983314734750475917808, −7.33949673377350984114135221936, −7.31718229705467604282642512916, −7.12455356976521705168102233679, −6.70967930478494331954492368715, −6.42390502245169325738949999789, −5.96817014498255965472221645776, −5.87451830210192037102660309192, −5.34921479461904539665305353664, −5.14900490916677743711880298576, −5.02276552586527159109362658071, −5.01580028698192399463106682734, −4.28613190554764124305030702875, −4.05843785717049740458989508376, −3.80937678388884616135425388614, −3.46003136619552125626240048643, −3.13083477174381427826509734249, −2.80424945573474713793001793859, −2.76272799281168445919596842905, −2.65773487301925712294831097862, −2.26605392336515945470032454479, −1.69802125686951890849849023551, −1.18017536120994828458719220230, −1.11309280645212246227281576968,
1.11309280645212246227281576968, 1.18017536120994828458719220230, 1.69802125686951890849849023551, 2.26605392336515945470032454479, 2.65773487301925712294831097862, 2.76272799281168445919596842905, 2.80424945573474713793001793859, 3.13083477174381427826509734249, 3.46003136619552125626240048643, 3.80937678388884616135425388614, 4.05843785717049740458989508376, 4.28613190554764124305030702875, 5.01580028698192399463106682734, 5.02276552586527159109362658071, 5.14900490916677743711880298576, 5.34921479461904539665305353664, 5.87451830210192037102660309192, 5.96817014498255965472221645776, 6.42390502245169325738949999789, 6.70967930478494331954492368715, 7.12455356976521705168102233679, 7.31718229705467604282642512916, 7.33949673377350984114135221936, 7.66455122983314734750475917808, 8.022779928948390959193001048335
