L(s) = 1 | + 2·3-s + 6·5-s + 20·7-s − 23·9-s + 14·11-s + 54·13-s + 12·15-s − 66·17-s + 162·19-s + 40·21-s + 172·23-s − 89·25-s − 100·27-s − 2·29-s − 128·31-s + 28·33-s + 120·35-s + 158·37-s + 108·39-s + 202·41-s − 298·43-s − 138·45-s − 408·47-s + 57·49-s − 132·51-s − 690·53-s + 84·55-s + ⋯ |
L(s) = 1 | + 0.384·3-s + 0.536·5-s + 1.07·7-s − 0.851·9-s + 0.383·11-s + 1.15·13-s + 0.206·15-s − 0.941·17-s + 1.95·19-s + 0.415·21-s + 1.55·23-s − 0.711·25-s − 0.712·27-s − 0.0128·29-s − 0.741·31-s + 0.147·33-s + 0.579·35-s + 0.702·37-s + 0.443·39-s + 0.769·41-s − 1.05·43-s − 0.457·45-s − 1.26·47-s + 0.166·49-s − 0.362·51-s − 1.78·53-s + 0.205·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.188354228\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.188354228\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 2 T + p^{3} T^{2} \) |
| 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 - 14 T + p^{3} T^{2} \) |
| 13 | \( 1 - 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 162 T + p^{3} T^{2} \) |
| 23 | \( 1 - 172 T + p^{3} T^{2} \) |
| 29 | \( 1 + 2 T + p^{3} T^{2} \) |
| 31 | \( 1 + 128 T + p^{3} T^{2} \) |
| 37 | \( 1 - 158 T + p^{3} T^{2} \) |
| 41 | \( 1 - 202 T + p^{3} T^{2} \) |
| 43 | \( 1 + 298 T + p^{3} T^{2} \) |
| 47 | \( 1 + 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 690 T + p^{3} T^{2} \) |
| 59 | \( 1 + 322 T + p^{3} T^{2} \) |
| 61 | \( 1 + 298 T + p^{3} T^{2} \) |
| 67 | \( 1 - 202 T + p^{3} T^{2} \) |
| 71 | \( 1 + 700 T + p^{3} T^{2} \) |
| 73 | \( 1 + 418 T + p^{3} T^{2} \) |
| 79 | \( 1 - 744 T + p^{3} T^{2} \) |
| 83 | \( 1 + 678 T + p^{3} T^{2} \) |
| 89 | \( 1 + 82 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1122 T + p^{3} T^{2} \) |
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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
−13.15734208924327732132487782521, −11.46308104391789043116989028070, −11.15488223876998859763639480538, −9.486920920769330789629509644938, −8.698950739839212579617931199420, −7.62042199960706018804023882758, −6.10652228390357149570939758253, −4.94873945156041834339812321202, −3.22236332720423801268702320469, −1.52048821417793549296260366590,
1.52048821417793549296260366590, 3.22236332720423801268702320469, 4.94873945156041834339812321202, 6.10652228390357149570939758253, 7.62042199960706018804023882758, 8.698950739839212579617931199420, 9.486920920769330789629509644938, 11.15488223876998859763639480538, 11.46308104391789043116989028070, 13.15734208924327732132487782521