| L(s) = 1 | + 4-s − 4·5-s − 4·11-s + 16-s − 4·20-s + 2·25-s − 4·37-s − 4·44-s + 16·47-s + 49-s + 4·53-s + 16·55-s + 64-s + 8·67-s − 16·71-s − 4·80-s − 20·89-s + 20·97-s + 2·100-s + 16·103-s − 4·113-s + 5·121-s + 28·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.78·5-s − 1.20·11-s + 1/4·16-s − 0.894·20-s + 2/5·25-s − 0.657·37-s − 0.603·44-s + 2.33·47-s + 1/7·49-s + 0.549·53-s + 2.15·55-s + 1/8·64-s + 0.977·67-s − 1.89·71-s − 0.447·80-s − 2.11·89-s + 2.03·97-s + 1/5·100-s + 1.57·103-s − 0.376·113-s + 5/11·121-s + 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920996 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.43688576794607557988711296474, −7.31514859453603203711352983098, −7.02933382590835884259254059518, −6.22587547680610001337765417218, −5.87227195463546443421041435616, −5.40884203814189531687833895140, −4.92858494933775383941758383647, −4.31149838803047807551804272396, −4.02562078900619730126988001703, −3.51596153015514902647191233740, −3.01214646522921378563614029232, −2.46861569775295984472920589718, −1.87413594762694485161682676302, −0.805025944192473153384098433748, 0,
0.805025944192473153384098433748, 1.87413594762694485161682676302, 2.46861569775295984472920589718, 3.01214646522921378563614029232, 3.51596153015514902647191233740, 4.02562078900619730126988001703, 4.31149838803047807551804272396, 4.92858494933775383941758383647, 5.40884203814189531687833895140, 5.87227195463546443421041435616, 6.22587547680610001337765417218, 7.02933382590835884259254059518, 7.31514859453603203711352983098, 7.43688576794607557988711296474
