| L(s) = 1 | + 4·7-s − 4·19-s + 7·25-s − 12·29-s − 20·31-s − 22·37-s − 18·47-s + 9·49-s + 24·53-s − 6·59-s − 30·83-s − 8·103-s + 10·109-s + 36·113-s + 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 14·169-s + 173-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.917·19-s + 7/5·25-s − 2.22·29-s − 3.59·31-s − 3.61·37-s − 2.62·47-s + 9/7·49-s + 3.29·53-s − 0.781·59-s − 3.29·83-s − 0.788·103-s + 0.957·109-s + 3.38·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-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 + 1.07·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.245626230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.245626230\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 79 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
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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
−8.811972540626731772952779411353, −8.647074114512688486035137679657, −8.330648532192857778115414626618, −7.73036194167570246420712176758, −7.19035527388077362313746820214, −7.08284676861751351858076882137, −7.07659731465896497768463635599, −6.18156199397365544615530035232, −5.66249617915409273086294898323, −5.44821123705890341560842717593, −5.09378139017334053161677800056, −4.80639147457528683836584717368, −4.15595609768175443976298305622, −3.73197264380886513699184592884, −3.50259086024534713934440278145, −2.87233602967682451436281993917, −1.91239697986041212921589527643, −1.85912610215612717847088602238, −1.56376177442796804589164893808, −0.32807845338339505349024143005,
0.32807845338339505349024143005, 1.56376177442796804589164893808, 1.85912610215612717847088602238, 1.91239697986041212921589527643, 2.87233602967682451436281993917, 3.50259086024534713934440278145, 3.73197264380886513699184592884, 4.15595609768175443976298305622, 4.80639147457528683836584717368, 5.09378139017334053161677800056, 5.44821123705890341560842717593, 5.66249617915409273086294898323, 6.18156199397365544615530035232, 7.07659731465896497768463635599, 7.08284676861751351858076882137, 7.19035527388077362313746820214, 7.73036194167570246420712176758, 8.330648532192857778115414626618, 8.647074114512688486035137679657, 8.811972540626731772952779411353
