| L(s) = 1 | + 3·5-s + 4·7-s − 3·9-s − 3·11-s − 5·13-s − 3·17-s + 5·19-s − 3·23-s + 5·25-s + 3·29-s + 8·31-s + 12·35-s + 7·37-s + 9·41-s + 11·43-s − 9·45-s + 9·49-s + 3·53-s − 9·55-s − 24·59-s + 4·61-s − 12·63-s − 15·65-s + 8·67-s − 11·73-s − 12·77-s − 16·79-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.51·7-s − 9-s − 0.904·11-s − 1.38·13-s − 0.727·17-s + 1.14·19-s − 0.625·23-s + 25-s + 0.557·29-s + 1.43·31-s + 2.02·35-s + 1.15·37-s + 1.40·41-s + 1.67·43-s − 1.34·45-s + 9/7·49-s + 0.412·53-s − 1.21·55-s − 3.12·59-s + 0.512·61-s − 1.51·63-s − 1.86·65-s + 0.977·67-s − 1.28·73-s − 1.36·77-s − 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.691204233\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.691204233\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 9 T + 40 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 11 T + 78 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3 T - 44 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−10.33798254719065105205939130188, −9.619263558612829866938456291901, −9.389423481983027939106977063499, −9.117083148226521486289176463372, −8.292487610769629102975359447154, −8.211533024844988270910848156246, −7.75988622186440004841201124106, −7.28640552385002937919133801497, −6.90872700278033865757392022791, −5.97982032825057503297626016508, −5.91556102688506950653119334527, −5.53266593559791639613374244290, −4.95060126155134455238889438821, −4.48488406827895221351525110804, −4.37568761540743302055633908492, −2.98385214526719550260434113981, −2.72201376125639236901541679403, −2.32138878337855543423008331692, −1.67499453810831390846075392936, −0.75348583842150611768694550139,
0.75348583842150611768694550139, 1.67499453810831390846075392936, 2.32138878337855543423008331692, 2.72201376125639236901541679403, 2.98385214526719550260434113981, 4.37568761540743302055633908492, 4.48488406827895221351525110804, 4.95060126155134455238889438821, 5.53266593559791639613374244290, 5.91556102688506950653119334527, 5.97982032825057503297626016508, 6.90872700278033865757392022791, 7.28640552385002937919133801497, 7.75988622186440004841201124106, 8.211533024844988270910848156246, 8.292487610769629102975359447154, 9.117083148226521486289176463372, 9.389423481983027939106977063499, 9.619263558612829866938456291901, 10.33798254719065105205939130188
