| L(s) = 1 | + 4·4-s + 12·16-s + 10·25-s + 26·43-s + 13·49-s − 26·61-s + 32·64-s − 26·79-s + 40·100-s + 26·103-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 104·172-s + 173-s + 179-s + 181-s + 191-s + 193-s + 52·196-s + ⋯ |
| L(s) = 1 | + 2·4-s + 3·16-s + 2·25-s + 3.96·43-s + 13/7·49-s − 3.32·61-s + 4·64-s − 2.92·79-s + 4·100-s + 2.56·103-s + 2·121-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 + 7.92·172-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 26/7·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.900934956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.900934956\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 143 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 169 T^{2} + p^{2} T^{4} \) |
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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
−9.659082610768094433200294304598, −9.295072359919803899908115580499, −8.684226119232464045011621180990, −8.641084654015650009378405767515, −7.85110923973955744832581489947, −7.45384282831554294061437511132, −7.19538398166026916977415403049, −7.12396746697999620413722732723, −6.26536375076026176873066859954, −6.11660382041295657273521805962, −5.77147114024431785919914255069, −5.31671394149868535975107086300, −4.47155518430251494812864844911, −4.32970111751942568956851864316, −3.36033846437619319020478767115, −3.13743671552165591930356222847, −2.44074225222256654961317362096, −2.33861750099995445869561081693, −1.35067390714073220600406077605, −0.951656736640315526814882379559,
0.951656736640315526814882379559, 1.35067390714073220600406077605, 2.33861750099995445869561081693, 2.44074225222256654961317362096, 3.13743671552165591930356222847, 3.36033846437619319020478767115, 4.32970111751942568956851864316, 4.47155518430251494812864844911, 5.31671394149868535975107086300, 5.77147114024431785919914255069, 6.11660382041295657273521805962, 6.26536375076026176873066859954, 7.12396746697999620413722732723, 7.19538398166026916977415403049, 7.45384282831554294061437511132, 7.85110923973955744832581489947, 8.641084654015650009378405767515, 8.684226119232464045011621180990, 9.295072359919803899908115580499, 9.659082610768094433200294304598
