| L(s) = 1 | + 7-s + 2·9-s + 8·23-s − 6·25-s + 4·29-s + 4·37-s + 49-s − 4·53-s + 2·63-s + 24·67-s − 16·71-s − 5·81-s + 8·107-s + 4·109-s − 12·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 8·161-s + 163-s + 167-s − 6·169-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 2/3·9-s + 1.66·23-s − 6/5·25-s + 0.742·29-s + 0.657·37-s + 1/7·49-s − 0.549·53-s + 0.251·63-s + 2.93·67-s − 1.89·71-s − 5/9·81-s + 0.773·107-s + 0.383·109-s − 1.12·113-s − 0.545·121-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 + 0.630·161-s + 0.0783·163-s + 0.0773·167-s − 0.461·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.479234028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.479234028\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 114 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
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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
−10.21153972567606006133163866615, −9.654094699996107964187743515989, −9.276950853486320200719383252542, −8.615872896780573684178537844856, −8.131359784741589074750371519071, −7.57167178525341996409762836293, −7.05393749937669474294137599816, −6.54626652354326609342532266340, −5.85440530134450522286067548295, −5.18544778512776051224363649089, −4.62178247274018777005148327946, −4.00703032339570304850982933014, −3.18534496418189824470981855562, −2.29527196011732596467924605502, −1.21195671673507740852473807590,
1.21195671673507740852473807590, 2.29527196011732596467924605502, 3.18534496418189824470981855562, 4.00703032339570304850982933014, 4.62178247274018777005148327946, 5.18544778512776051224363649089, 5.85440530134450522286067548295, 6.54626652354326609342532266340, 7.05393749937669474294137599816, 7.57167178525341996409762836293, 8.131359784741589074750371519071, 8.615872896780573684178537844856, 9.276950853486320200719383252542, 9.654094699996107964187743515989, 10.21153972567606006133163866615
