| L(s) = 1 | − 7-s + 6·11-s − 2·23-s − 6·25-s + 4·29-s + 8·37-s + 14·43-s + 49-s + 6·53-s + 8·67-s + 16·71-s − 6·77-s + 12·79-s − 9·81-s + 4·107-s + 4·109-s − 8·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 2·161-s + 163-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.80·11-s − 0.417·23-s − 6/5·25-s + 0.742·29-s + 1.31·37-s + 2.13·43-s + 1/7·49-s + 0.824·53-s + 0.977·67-s + 1.89·71-s − 0.683·77-s + 1.35·79-s − 81-s + 0.386·107-s + 0.383·109-s − 0.752·113-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.157·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.362058470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.362058470\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 96 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
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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
−7.973154762079545303604438435230, −7.53501780426551273067960397429, −6.96896607652998296250929779979, −6.63945337322056470396257157759, −6.19922578082538972545274879674, −5.90477584666526868227975885217, −5.40533381932107717723736343107, −4.72351915570667030998933327870, −4.19721540899933601187892140421, −3.84837293664482043821669858618, −3.54867190725794325861079800664, −2.60179529300555599455104946686, −2.26861390332467851560893936650, −1.36299850813920827589267426179, −0.72965766052278370661407714693,
0.72965766052278370661407714693, 1.36299850813920827589267426179, 2.26861390332467851560893936650, 2.60179529300555599455104946686, 3.54867190725794325861079800664, 3.84837293664482043821669858618, 4.19721540899933601187892140421, 4.72351915570667030998933327870, 5.40533381932107717723736343107, 5.90477584666526868227975885217, 6.19922578082538972545274879674, 6.63945337322056470396257157759, 6.96896607652998296250929779979, 7.53501780426551273067960397429, 7.973154762079545303604438435230
