| L(s) = 1 | − 2·3-s + 6·7-s + 2·9-s − 12·21-s + 2·23-s − 6·27-s + 12·29-s − 18·43-s + 14·47-s + 18·49-s + 12·63-s − 6·67-s − 4·69-s + 11·81-s + 22·83-s − 24·87-s + 12·89-s + 36·101-s + 18·103-s + 26·107-s − 22·121-s + 127-s + 36·129-s + 131-s + 137-s + 139-s − 28·141-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 2.26·7-s + 2/3·9-s − 2.61·21-s + 0.417·23-s − 1.15·27-s + 2.22·29-s − 2.74·43-s + 2.04·47-s + 18/7·49-s + 1.51·63-s − 0.733·67-s − 0.481·69-s + 11/9·81-s + 2.41·83-s − 2.57·87-s + 1.27·89-s + 3.58·101-s + 1.77·103-s + 2.51·107-s − 2·121-s + 0.0887·127-s + 3.16·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.35·141-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.159837556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.159837556\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
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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
−9.808634071172554708725211779728, −8.981021072051790332711909896389, −8.715682100582172526057577956387, −8.553190499172138405910183511041, −7.86922279576880230218534033104, −7.62561266042530916678711360244, −7.35729692236798593523829620330, −6.72016697862793868423748699054, −6.17364264952357770932618056764, −6.10430167146131886553183041943, −5.29369115774265646522695955691, −5.06187783798438436590623467379, −4.69630400834468146692411967977, −4.55449588525619354671206126566, −3.72895650149875186242656031563, −3.27386988931851935633473820789, −2.31422058439203483863311661110, −1.96314886680260062258064664354, −1.24744484253273168861608210071, −0.69687543080154082339848196756,
0.69687543080154082339848196756, 1.24744484253273168861608210071, 1.96314886680260062258064664354, 2.31422058439203483863311661110, 3.27386988931851935633473820789, 3.72895650149875186242656031563, 4.55449588525619354671206126566, 4.69630400834468146692411967977, 5.06187783798438436590623467379, 5.29369115774265646522695955691, 6.10430167146131886553183041943, 6.17364264952357770932618056764, 6.72016697862793868423748699054, 7.35729692236798593523829620330, 7.62561266042530916678711360244, 7.86922279576880230218534033104, 8.553190499172138405910183511041, 8.715682100582172526057577956387, 8.981021072051790332711909896389, 9.808634071172554708725211779728
