| L(s) = 1 | + 4·3-s + 10·9-s + 8·19-s − 2·25-s + 20·27-s − 8·29-s − 24·31-s + 8·37-s + 24·47-s + 10·49-s + 32·53-s + 32·57-s − 8·75-s + 35·81-s + 8·83-s − 32·87-s − 96·93-s − 16·103-s − 16·109-s + 32·111-s + 16·113-s + 32·121-s + 127-s + 131-s + 137-s + 139-s + 96·141-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 10/3·9-s + 1.83·19-s − 2/5·25-s + 3.84·27-s − 1.48·29-s − 4.31·31-s + 1.31·37-s + 3.50·47-s + 10/7·49-s + 4.39·53-s + 4.23·57-s − 0.923·75-s + 35/9·81-s + 0.878·83-s − 3.43·87-s − 9.95·93-s − 1.57·103-s − 1.53·109-s + 3.03·111-s + 1.50·113-s + 2.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 8.08·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.91293021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.91293021\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 498 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 12 T^{2} + 230 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 12 T + 92 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 84 T^{2} + 3590 T^{4} - 84 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 16 T + 164 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 94 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 2454 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 4842 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 240 T^{2} + 24098 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 176 T^{2} + 17538 T^{4} - 176 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 146 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 92 T^{2} + 4134 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 80 T^{2} - 1182 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.02979231427515908749839920800, −6.38026233537271728319648565137, −6.37932063148259777997440768714, −5.86363962019068917424674731006, −5.72067153688330870758177989927, −5.46793899245093913818527163091, −5.44363323381552612792737485364, −5.34151734150130545561819480972, −4.98064491375980754541589763488, −4.47415733994069151318969832395, −4.21041731758539221966250238338, −4.12599431207327027800608773568, −3.85164459909305969337897632948, −3.78156751335638955463648125158, −3.55417028238799648832244519310, −3.19642820142914843569357855982, −3.15444597303540270348037309263, −2.45497332384701383679413938179, −2.42495557116120807989579094508, −2.39393546353079963552168374081, −2.04910447061723562425927026940, −1.59020191859825731891739362254, −1.39731003537580246177117638474, −0.834850455633638483761216298867, −0.57669614586899238532430541941,
0.57669614586899238532430541941, 0.834850455633638483761216298867, 1.39731003537580246177117638474, 1.59020191859825731891739362254, 2.04910447061723562425927026940, 2.39393546353079963552168374081, 2.42495557116120807989579094508, 2.45497332384701383679413938179, 3.15444597303540270348037309263, 3.19642820142914843569357855982, 3.55417028238799648832244519310, 3.78156751335638955463648125158, 3.85164459909305969337897632948, 4.12599431207327027800608773568, 4.21041731758539221966250238338, 4.47415733994069151318969832395, 4.98064491375980754541589763488, 5.34151734150130545561819480972, 5.44363323381552612792737485364, 5.46793899245093913818527163091, 5.72067153688330870758177989927, 5.86363962019068917424674731006, 6.37932063148259777997440768714, 6.38026233537271728319648565137, 7.02979231427515908749839920800
