| L(s) = 1 | + 4·3-s − 8·7-s + 4·9-s + 40·13-s + 32·19-s − 32·21-s − 10·25-s + 4·27-s − 32·31-s + 88·37-s + 160·39-s + 56·43-s + 24·49-s + 128·57-s + 64·61-s − 32·63-s − 328·67-s + 200·73-s − 40·75-s + 112·79-s + 31·81-s − 320·91-s − 128·93-s + 296·97-s − 296·103-s − 80·109-s + 352·111-s + ⋯ |
| L(s) = 1 | + 4/3·3-s − 8/7·7-s + 4/9·9-s + 3.07·13-s + 1.68·19-s − 1.52·21-s − 2/5·25-s + 4/27·27-s − 1.03·31-s + 2.37·37-s + 4.10·39-s + 1.30·43-s + 0.489·49-s + 2.24·57-s + 1.04·61-s − 0.507·63-s − 4.89·67-s + 2.73·73-s − 0.533·75-s + 1.41·79-s + 0.382·81-s − 3.51·91-s − 1.37·93-s + 3.05·97-s − 2.87·103-s − 0.733·109-s + 3.17·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(9.258090565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.258090565\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 4 T + 4 p T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| good | 7 | $D_{4}$ | \( ( 1 + 4 T + 12 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{4} \) |
| 17 | $D_4\times C_2$ | \( 1 - 220 T^{2} - 28218 T^{4} - 220 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 16 T + 426 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 1720 T^{2} + 1286322 T^{4} - 1720 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 962 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 - 44 T + 1782 T^{2} - 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 4060 T^{2} + 8942982 T^{4} - 4060 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 28 T + 3084 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 1064 T^{2} + 1942386 T^{4} + 1064 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 10300 T^{2} + 42096102 T^{4} - 10300 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 7300 T^{2} + 30092262 T^{4} - 7300 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 32 T + 6258 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 164 T + 14892 T^{2} + 164 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 15124 T^{2} + 102823206 T^{4} - 15124 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 100 T + 11718 T^{2} - 100 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 56 T + 12906 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 26872 T^{2} + 275326098 T^{4} - 26872 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 27940 T^{2} + 317327622 T^{4} - 27940 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 148 T + 22854 T^{2} - 148 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.03401539057959107411182036438, −6.88670437578468016014561942727, −6.24540600068557187937570186181, −6.23245377410674315847483329622, −6.17418003666519197271765918865, −6.00548593016296139782025689758, −5.72411601993127334564291905652, −5.43323522189182917926519060622, −5.26674870922384965221467466953, −4.73995269754208970589588087759, −4.56325354291431779951835249928, −4.23516664499465729506060221591, −4.12418253962500591955009480897, −3.55033609488002054732154762235, −3.48026193960884924941621258264, −3.36162972751559906463752608244, −3.35291963688369324165529618072, −2.71887408378686932376456410469, −2.62682297773042309014523442617, −2.26666734807404621267417377790, −1.88054806960628135595630255889, −1.43096046073162840379416432940, −1.07785550114039039955309499393, −0.877239607048254601442338705520, −0.41303600885086426766623521300,
0.41303600885086426766623521300, 0.877239607048254601442338705520, 1.07785550114039039955309499393, 1.43096046073162840379416432940, 1.88054806960628135595630255889, 2.26666734807404621267417377790, 2.62682297773042309014523442617, 2.71887408378686932376456410469, 3.35291963688369324165529618072, 3.36162972751559906463752608244, 3.48026193960884924941621258264, 3.55033609488002054732154762235, 4.12418253962500591955009480897, 4.23516664499465729506060221591, 4.56325354291431779951835249928, 4.73995269754208970589588087759, 5.26674870922384965221467466953, 5.43323522189182917926519060622, 5.72411601993127334564291905652, 6.00548593016296139782025689758, 6.17418003666519197271765918865, 6.23245377410674315847483329622, 6.24540600068557187937570186181, 6.88670437578468016014561942727, 7.03401539057959107411182036438
