| L(s) = 1 | + 5-s + 4·7-s + 4·11-s + 2·13-s − 4·17-s + 8·19-s + 4·23-s − 2·29-s + 8·31-s + 4·35-s + 12·37-s − 6·41-s + 8·43-s + 4·47-s + 7·49-s − 12·53-s + 4·55-s − 4·59-s + 2·61-s + 2·65-s − 8·67-s − 12·73-s + 16·77-s − 16·83-s − 4·85-s + 12·89-s + 8·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 1.20·11-s + 0.554·13-s − 0.970·17-s + 1.83·19-s + 0.834·23-s − 0.371·29-s + 1.43·31-s + 0.676·35-s + 1.97·37-s − 0.937·41-s + 1.21·43-s + 0.583·47-s + 49-s − 1.64·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s − 1.40·73-s + 1.82·77-s − 1.75·83-s − 0.433·85-s + 1.27·89-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.434640946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.434640946\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 8 T - 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 16 T + 173 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−8.775738221971129928713017847475, −8.620925990908311200375000500775, −8.055497835935780895990818239749, −7.70580819100260057630002073703, −7.39564614325142566789176250416, −7.10097507023005863153467861656, −6.45354592331089491291581107900, −6.25473549548713818342062133531, −5.79283828259738763106229961254, −5.52935531655825512705437523890, −4.77255057291672011978712871596, −4.66002107074911457044767329640, −4.39401867612762282632330780686, −3.79866559309019276402115184434, −3.14499468958761546637218678166, −2.92894620361717558564372310535, −2.16028693128308461113817397367, −1.71580801263595136428293652893, −1.15445220581255516115920080329, −0.855327904736472794737847602444,
0.855327904736472794737847602444, 1.15445220581255516115920080329, 1.71580801263595136428293652893, 2.16028693128308461113817397367, 2.92894620361717558564372310535, 3.14499468958761546637218678166, 3.79866559309019276402115184434, 4.39401867612762282632330780686, 4.66002107074911457044767329640, 4.77255057291672011978712871596, 5.52935531655825512705437523890, 5.79283828259738763106229961254, 6.25473549548713818342062133531, 6.45354592331089491291581107900, 7.10097507023005863153467861656, 7.39564614325142566789176250416, 7.70580819100260057630002073703, 8.055497835935780895990818239749, 8.620925990908311200375000500775, 8.775738221971129928713017847475
