| L(s) = 1 | − 1.07·3-s + 11.6·5-s + 2.67·7-s − 25.8·9-s + 63.9·11-s + 50.0·13-s − 12.4·15-s + 72.4·17-s + 27.4·19-s − 2.85·21-s − 139.·23-s + 10.3·25-s + 56.5·27-s − 93.3·29-s + 188.·31-s − 68.4·33-s + 31.0·35-s − 118.·37-s − 53.5·39-s + 104.·41-s + 44.5·43-s − 300.·45-s + 488.·47-s − 335.·49-s − 77.5·51-s − 211.·53-s + 743.·55-s + ⋯ |
| L(s) = 1 | − 0.205·3-s + 1.04·5-s + 0.144·7-s − 0.957·9-s + 1.75·11-s + 1.06·13-s − 0.214·15-s + 1.03·17-s + 0.332·19-s − 0.0297·21-s − 1.26·23-s + 0.0826·25-s + 0.403·27-s − 0.598·29-s + 1.09·31-s − 0.361·33-s + 0.150·35-s − 0.528·37-s − 0.219·39-s + 0.398·41-s + 0.157·43-s − 0.996·45-s + 1.51·47-s − 0.979·49-s − 0.213·51-s − 0.548·53-s + 1.82·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.804765707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.804765707\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 1.07T + 27T^{2} \) |
| 5 | \( 1 - 11.6T + 125T^{2} \) |
| 7 | \( 1 - 2.67T + 343T^{2} \) |
| 11 | \( 1 - 63.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 50.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 72.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 139.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 93.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 188.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 118.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 104.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 44.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 488.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 211.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 402.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 322.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 196.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 453.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 259.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 323.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 797.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 866.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 936.T + 9.12e5T^{2} \) |
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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
−9.566449358293253371526906330906, −8.826819928214453141513089501983, −8.067989637357565257535457190789, −6.79124003860099910583007562232, −5.93751272540460068504086959115, −5.66432938630262054469243703483, −4.19651908209418389351757072040, −3.26744522557501404478961796726, −1.90377093689440825680717598967, −0.962157248111783288151881205879,
0.962157248111783288151881205879, 1.90377093689440825680717598967, 3.26744522557501404478961796726, 4.19651908209418389351757072040, 5.66432938630262054469243703483, 5.93751272540460068504086959115, 6.79124003860099910583007562232, 8.067989637357565257535457190789, 8.826819928214453141513089501983, 9.566449358293253371526906330906
