L(s) = 1 | − 5-s + 11-s − 3·13-s + 2·17-s + 2·19-s − 2·23-s + 25-s + 9·31-s + 10·37-s + 12·41-s − 43-s − 10·47-s + 14·53-s − 55-s − 9·59-s − 14·61-s + 3·65-s + 10·67-s + 5·71-s + 11·73-s + 10·79-s + 9·83-s − 2·85-s − 89-s − 2·95-s − 6·97-s + 101-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.301·11-s − 0.832·13-s + 0.485·17-s + 0.458·19-s − 0.417·23-s + 1/5·25-s + 1.61·31-s + 1.64·37-s + 1.87·41-s − 0.152·43-s − 1.45·47-s + 1.92·53-s − 0.134·55-s − 1.17·59-s − 1.79·61-s + 0.372·65-s + 1.22·67-s + 0.593·71-s + 1.28·73-s + 1.12·79-s + 0.987·83-s − 0.216·85-s − 0.105·89-s − 0.205·95-s − 0.609·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.545646850\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.545646850\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.90542081890586, −13.31884040651176, −12.67817800316051, −12.26130668217537, −11.93679927407357, −11.29288962106297, −11.02142146406034, −10.17901625908874, −9.871478735911547, −9.379439420819376, −8.866040150539856, −8.102184608179015, −7.746216856299807, −7.451833579649175, −6.569045016643164, −6.273197238964313, −5.616116920351327, −4.878900699541370, −4.543338098560337, −3.893302436994915, −3.264618458278880, −2.639354079058628, −2.096109814530243, −1.062961920708135, −0.5912161975841336,
0.5912161975841336, 1.062961920708135, 2.096109814530243, 2.639354079058628, 3.264618458278880, 3.893302436994915, 4.543338098560337, 4.878900699541370, 5.616116920351327, 6.273197238964313, 6.569045016643164, 7.451833579649175, 7.746216856299807, 8.102184608179015, 8.866040150539856, 9.379439420819376, 9.871478735911547, 10.17901625908874, 11.02142146406034, 11.29288962106297, 11.93679927407357, 12.26130668217537, 12.67817800316051, 13.31884040651176, 13.90542081890586