| L(s) = 1 | − 3-s + 3·5-s + 3·9-s − 8·11-s − 5·13-s − 3·15-s + 5·17-s + 8·19-s − 23-s + 5·25-s − 8·27-s + 3·29-s − 8·31-s + 8·33-s − 4·37-s + 5·39-s + 5·41-s + 11·43-s + 9·45-s − 5·47-s − 14·49-s − 5·51-s − 9·53-s − 24·55-s − 8·57-s − 13·59-s − 61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 9-s − 2.41·11-s − 1.38·13-s − 0.774·15-s + 1.21·17-s + 1.83·19-s − 0.208·23-s + 25-s − 1.53·27-s + 0.557·29-s − 1.43·31-s + 1.39·33-s − 0.657·37-s + 0.800·39-s + 0.780·41-s + 1.67·43-s + 1.34·45-s − 0.729·47-s − 2·49-s − 0.700·51-s − 1.23·53-s − 3.23·55-s − 1.05·57-s − 1.69·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.659727651\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.659727651\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 5 T + 8 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 5 T - 22 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 9 T + 28 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 13 T + 110 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - T - 70 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 9 T + 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.715180820042853961624359963375, −9.640916440371295828170286907160, −9.580556452842616133874707824549, −8.895873358642018484383126156128, −7.950334559990657000152670075033, −7.80740443640755218767881545153, −7.44493612495742036205885590942, −7.35394018323993776704115904559, −6.46276062270330181950929496883, −6.06109566245173621023143233127, −5.71773145201637802273942533246, −5.01657411371967860326931911384, −4.98646779021239085606869713691, −4.94281694520370834959707764673, −3.63621842988232333853591875633, −3.28109446348280584968007258376, −2.60821114431416979771409002139, −2.12525329649672895441203633233, −1.56284473853429568989784086135, −0.56279527096814167375525697871,
0.56279527096814167375525697871, 1.56284473853429568989784086135, 2.12525329649672895441203633233, 2.60821114431416979771409002139, 3.28109446348280584968007258376, 3.63621842988232333853591875633, 4.94281694520370834959707764673, 4.98646779021239085606869713691, 5.01657411371967860326931911384, 5.71773145201637802273942533246, 6.06109566245173621023143233127, 6.46276062270330181950929496883, 7.35394018323993776704115904559, 7.44493612495742036205885590942, 7.80740443640755218767881545153, 7.950334559990657000152670075033, 8.895873358642018484383126156128, 9.580556452842616133874707824549, 9.640916440371295828170286907160, 9.715180820042853961624359963375
