L(s) = 1 | + 7-s + 2·9-s + 2·11-s − 10·23-s + 8·25-s + 2·29-s + 14·37-s − 10·43-s + 49-s − 12·53-s + 2·63-s − 8·67-s + 8·71-s + 2·77-s − 4·79-s − 5·81-s + 4·99-s + 8·107-s + 26·109-s + 20·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 2/3·9-s + 0.603·11-s − 2.08·23-s + 8/5·25-s + 0.371·29-s + 2.30·37-s − 1.52·43-s + 1/7·49-s − 1.64·53-s + 0.251·63-s − 0.977·67-s + 0.949·71-s + 0.227·77-s − 0.450·79-s − 5/9·81-s + 0.402·99-s + 0.773·107-s + 2.49·109-s + 1.88·113-s − 0.909·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404928 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.418855533\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.418855533\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
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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
−7.956664309442800493331047630212, −7.58859647739024456683706443913, −7.05010148801106154077244098451, −6.63816887714249518144690975133, −6.14134670929552779051308300303, −5.98993005102216984812083460162, −5.19533750864123085075422006196, −4.72369091466956213083366103735, −4.36796661297590203351328709610, −3.98942824240376064026147735468, −3.27601579230360099330992906539, −2.81990333719384775133769833012, −1.98868925761822168438950328796, −1.55249081098435501810838380656, −0.70304746349131344247647173726,
0.70304746349131344247647173726, 1.55249081098435501810838380656, 1.98868925761822168438950328796, 2.81990333719384775133769833012, 3.27601579230360099330992906539, 3.98942824240376064026147735468, 4.36796661297590203351328709610, 4.72369091466956213083366103735, 5.19533750864123085075422006196, 5.98993005102216984812083460162, 6.14134670929552779051308300303, 6.63816887714249518144690975133, 7.05010148801106154077244098451, 7.58859647739024456683706443913, 7.956664309442800493331047630212