| L(s) = 1 | − 4·5-s + 9-s + 4·11-s − 12·23-s + 11·25-s − 8·31-s − 8·37-s − 4·45-s + 4·47-s + 2·49-s + 12·53-s − 16·55-s − 8·59-s − 8·67-s + 81-s − 4·89-s + 4·99-s + 16·103-s − 32·113-s + 48·115-s + 5·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 1/3·9-s + 1.20·11-s − 2.50·23-s + 11/5·25-s − 1.43·31-s − 1.31·37-s − 0.596·45-s + 0.583·47-s + 2/7·49-s + 1.64·53-s − 2.15·55-s − 1.04·59-s − 0.977·67-s + 1/9·81-s − 0.423·89-s + 0.402·99-s + 1.57·103-s − 3.01·113-s + 4.47·115-s + 5/11·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1742400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8238872553\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8238872553\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.76180499734868177488991624268, −7.43804998480459994940256028008, −7.11978562847697108370287627441, −6.61290582660518014203971641778, −6.22918488752641509252490039069, −5.64884941463700543009802273063, −5.21188814196251173093077097831, −4.50360389450717499523447458231, −4.11280941964446380869377634811, −3.83273110547276971465030897409, −3.56566180884109152916089041146, −2.82380575648028662742275222434, −1.99273975832167782899006984368, −1.42623318383260889822751223240, −0.39144571532624511252138832324,
0.39144571532624511252138832324, 1.42623318383260889822751223240, 1.99273975832167782899006984368, 2.82380575648028662742275222434, 3.56566180884109152916089041146, 3.83273110547276971465030897409, 4.11280941964446380869377634811, 4.50360389450717499523447458231, 5.21188814196251173093077097831, 5.64884941463700543009802273063, 6.22918488752641509252490039069, 6.61290582660518014203971641778, 7.11978562847697108370287627441, 7.43804998480459994940256028008, 7.76180499734868177488991624268
