| L(s) = 1 | − 3·9-s − 10·11-s − 12·19-s + 18·29-s − 8·31-s − 8·41-s − 49-s + 16·59-s − 16·61-s + 16·71-s − 26·79-s − 8·89-s + 30·99-s + 12·101-s + 6·109-s + 53·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s + 36·171-s + ⋯ |
| L(s) = 1 | − 9-s − 3.01·11-s − 2.75·19-s + 3.34·29-s − 1.43·31-s − 1.24·41-s − 1/7·49-s + 2.08·59-s − 2.04·61-s + 1.89·71-s − 2.92·79-s − 0.847·89-s + 3.01·99-s + 1.19·101-s + 0.574·109-s + 4.81·121-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 + 0.0783·163-s + 0.0773·167-s + 1.30·169-s + 2.75·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4633195139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4633195139\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
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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
−10.59488606638497802590471713852, −10.28075442652587007118086495539, −10.15362406494797416618628475084, −9.372014338397061812541504654221, −8.554091312818740045830200041330, −8.489742183690883841737694427104, −8.290200500641146864512453704002, −7.80716655481365557359450062159, −7.11136986237823010309136103811, −6.73477479928210766509869206677, −6.16273023388888876329538437111, −5.68003026494937840684664637836, −5.29922882007952681476046861828, −4.66219609678196210872593875525, −4.49095532128702112856630399602, −3.49254032821134777410237607631, −2.77702407396729899599472009502, −2.58645479323628559486467999830, −1.93571542801500409457149809263, −0.33054285764930520896086152591,
0.33054285764930520896086152591, 1.93571542801500409457149809263, 2.58645479323628559486467999830, 2.77702407396729899599472009502, 3.49254032821134777410237607631, 4.49095532128702112856630399602, 4.66219609678196210872593875525, 5.29922882007952681476046861828, 5.68003026494937840684664637836, 6.16273023388888876329538437111, 6.73477479928210766509869206677, 7.11136986237823010309136103811, 7.80716655481365557359450062159, 8.290200500641146864512453704002, 8.489742183690883841737694427104, 8.554091312818740045830200041330, 9.372014338397061812541504654221, 10.15362406494797416618628475084, 10.28075442652587007118086495539, 10.59488606638497802590471713852
