| L(s) = 1 | + 5-s − 3.77·7-s − 3.77·11-s − 13-s − 3.77·17-s − 6·19-s − 1.77·23-s + 25-s − 2·29-s − 6·31-s − 3.77·35-s − 3.77·37-s + 0.227·41-s − 8·43-s + 6·47-s + 7.22·49-s − 3.77·53-s − 3.77·55-s + 13.5·59-s + 3.77·61-s − 65-s + 9.54·67-s + 15.3·71-s + 10·73-s + 14.2·77-s − 9.77·79-s − 9.54·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.42·7-s − 1.13·11-s − 0.277·13-s − 0.914·17-s − 1.37·19-s − 0.369·23-s + 0.200·25-s − 0.371·29-s − 1.07·31-s − 0.637·35-s − 0.620·37-s + 0.0356·41-s − 1.21·43-s + 0.875·47-s + 1.03·49-s − 0.518·53-s − 0.508·55-s + 1.76·59-s + 0.482·61-s − 0.124·65-s + 1.16·67-s + 1.81·71-s + 1.17·73-s + 1.62·77-s − 1.09·79-s − 1.04·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6304860250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6304860250\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 3.77T + 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 17 | \( 1 + 3.77T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 + 1.77T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 3.77T + 37T^{2} \) |
| 41 | \( 1 - 0.227T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 3.77T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 - 3.77T + 61T^{2} \) |
| 67 | \( 1 - 9.54T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 9.77T + 79T^{2} \) |
| 83 | \( 1 + 9.54T + 83T^{2} \) |
| 89 | \( 1 - 3.77T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.64272740334886364878886839364, −6.77797940268349296520128325108, −6.53275552318594001179211391517, −5.63557868092530569695604515297, −5.11057956709598084093072496904, −4.09949782251354347393943913365, −3.43915621066527651348112237008, −2.47845019996357048977976409115, −2.03751615440804739807679575461, −0.34917212579683962916964644732,
0.34917212579683962916964644732, 2.03751615440804739807679575461, 2.47845019996357048977976409115, 3.43915621066527651348112237008, 4.09949782251354347393943913365, 5.11057956709598084093072496904, 5.63557868092530569695604515297, 6.53275552318594001179211391517, 6.77797940268349296520128325108, 7.64272740334886364878886839364
