L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s + 11-s − 12-s + 13-s + 14-s + 16-s − 17-s + 18-s + 19-s − 21-s + 22-s + 23-s − 24-s + 26-s − 27-s + 28-s − 29-s − 31-s + 32-s − 33-s − 34-s + ⋯ |
L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 7-s + 8-s + 9-s + 11-s − 12-s + 13-s + 14-s + 16-s − 17-s + 18-s + 19-s − 21-s + 22-s + 23-s − 24-s + 26-s − 27-s + 28-s − 29-s − 31-s + 32-s − 33-s − 34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.861491889\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.861491889\) |
\(L(1)\) |
\(\approx\) |
\(1.850151876\) |
\(L(1)\) |
\(\approx\) |
\(1.850151876\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 229 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.594021540832812203608300682557, −20.60819841966646448304383607121, −20.18518788508950556052181767277, −18.90045823468060961155687190693, −18.15287471136325937467683938092, −17.19925551140539515945064484807, −16.73128493115027668697275427837, −15.69022861632903257289560347048, −15.18840772707070224274893247479, −14.17956112430566345006411740349, −13.49194182866692446159760467939, −12.638326117655462323508973723230, −11.76886804943861869444144496396, −11.168362286202846729801371291206, −10.84269812350252494026383413922, −9.50985933794485593006639543379, −8.40014973792814400852067859395, −7.1731990858836848329356889167, −6.717369843497139389909975801103, −5.65218444699319510114451448600, −5.10133533456546418237773331502, −4.19199431334497285465653490652, −3.465716185875986126005266238769, −1.85114040266492740490138226870, −1.245047312701778168427478892934,
1.245047312701778168427478892934, 1.85114040266492740490138226870, 3.465716185875986126005266238769, 4.19199431334497285465653490652, 5.10133533456546418237773331502, 5.65218444699319510114451448600, 6.717369843497139389909975801103, 7.1731990858836848329356889167, 8.40014973792814400852067859395, 9.50985933794485593006639543379, 10.84269812350252494026383413922, 11.168362286202846729801371291206, 11.76886804943861869444144496396, 12.638326117655462323508973723230, 13.49194182866692446159760467939, 14.17956112430566345006411740349, 15.18840772707070224274893247479, 15.69022861632903257289560347048, 16.73128493115027668697275427837, 17.19925551140539515945064484807, 18.15287471136325937467683938092, 18.90045823468060961155687190693, 20.18518788508950556052181767277, 20.60819841966646448304383607121, 21.594021540832812203608300682557