L(s) = 1 | − 3-s + 3·5-s + 9-s − 3·11-s − 4·13-s − 3·15-s + 4·19-s + 4·25-s − 27-s + 9·29-s + 31-s + 3·33-s + 8·37-s + 4·39-s + 10·43-s + 3·45-s + 6·47-s − 3·53-s − 9·55-s − 4·57-s − 3·59-s − 10·61-s − 12·65-s + 10·67-s + 6·71-s + 2·73-s − 4·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1.34·5-s + 1/3·9-s − 0.904·11-s − 1.10·13-s − 0.774·15-s + 0.917·19-s + 4/5·25-s − 0.192·27-s + 1.67·29-s + 0.179·31-s + 0.522·33-s + 1.31·37-s + 0.640·39-s + 1.52·43-s + 0.447·45-s + 0.875·47-s − 0.412·53-s − 1.21·55-s − 0.529·57-s − 0.390·59-s − 1.28·61-s − 1.48·65-s + 1.22·67-s + 0.712·71-s + 0.234·73-s − 0.461·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.770811931\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.770811931\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.349559860338236627307047199547, −8.101414765273314185762497375802, −7.40191914641952926424395037725, −6.50699481561569069437860761870, −5.79013911758710290848318889741, −5.16285734612725132936189698989, −4.47029016100817810999387000111, −2.89536498281435943334574976705, −2.23267398249755688928371513634, −0.890185389193627517801934971595,
0.890185389193627517801934971595, 2.23267398249755688928371513634, 2.89536498281435943334574976705, 4.47029016100817810999387000111, 5.16285734612725132936189698989, 5.79013911758710290848318889741, 6.50699481561569069437860761870, 7.40191914641952926424395037725, 8.101414765273314185762497375802, 9.349559860338236627307047199547