| L(s) = 1 | + 6.18·5-s + 22.9·7-s + 4.37·11-s − 70.0·13-s − 83.2·17-s + 158.·19-s − 23·23-s − 86.7·25-s + 25.4·29-s + 179.·31-s + 141.·35-s + 35.1·37-s + 209.·41-s + 439.·43-s + 276.·47-s + 182.·49-s − 236.·53-s + 27.0·55-s − 786.·59-s + 706.·61-s − 433.·65-s + 307.·67-s + 723.·71-s + 553.·73-s + 100.·77-s + 153.·79-s − 386.·83-s + ⋯ |
| L(s) = 1 | + 0.552·5-s + 1.23·7-s + 0.119·11-s − 1.49·13-s − 1.18·17-s + 1.91·19-s − 0.208·23-s − 0.694·25-s + 0.162·29-s + 1.04·31-s + 0.684·35-s + 0.156·37-s + 0.797·41-s + 1.56·43-s + 0.857·47-s + 0.532·49-s − 0.612·53-s + 0.0662·55-s − 1.73·59-s + 1.48·61-s − 0.826·65-s + 0.560·67-s + 1.20·71-s + 0.886·73-s + 0.148·77-s + 0.218·79-s − 0.510·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.776648008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.776648008\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 6.18T + 125T^{2} \) |
| 7 | \( 1 - 22.9T + 343T^{2} \) |
| 11 | \( 1 - 4.37T + 1.33e3T^{2} \) |
| 13 | \( 1 + 70.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 83.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 158.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 25.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 179.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 35.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 209.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 439.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 276.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 236.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 786.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 706.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 307.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 723.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 553.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 153.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 386.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 397.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.244164792882923032365301496727, −8.007753024027030015974131700705, −7.58933478412129392309926509405, −6.64521609847727747005991835222, −5.57969812961892755848886012113, −4.93939841232866133958399163186, −4.18250041455324463365333168118, −2.71085715341169064523269117222, −1.98068534471254867158603140772, −0.810473413013610028552218869443,
0.810473413013610028552218869443, 1.98068534471254867158603140772, 2.71085715341169064523269117222, 4.18250041455324463365333168118, 4.93939841232866133958399163186, 5.57969812961892755848886012113, 6.64521609847727747005991835222, 7.58933478412129392309926509405, 8.007753024027030015974131700705, 9.244164792882923032365301496727
