| L(s) = 1 | + 5·5-s − 10.3·7-s + 57.3·11-s − 22.3·13-s + 106.·17-s + 43.7·19-s − 16.4·23-s + 25·25-s + 57.5·29-s − 175.·31-s − 51.6·35-s − 280.·37-s − 157.·41-s + 457.·43-s + 18.6·47-s − 236.·49-s − 370.·53-s + 286.·55-s + 416.·59-s + 867.·61-s − 111.·65-s + 37.8·67-s − 83.6·71-s − 12.1·73-s − 592.·77-s + 175.·79-s + 572.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.558·7-s + 1.57·11-s − 0.475·13-s + 1.51·17-s + 0.528·19-s − 0.149·23-s + 0.200·25-s + 0.368·29-s − 1.01·31-s − 0.249·35-s − 1.24·37-s − 0.600·41-s + 1.62·43-s + 0.0578·47-s − 0.688·49-s − 0.960·53-s + 0.703·55-s + 0.918·59-s + 1.82·61-s − 0.212·65-s + 0.0690·67-s − 0.139·71-s − 0.0195·73-s − 0.877·77-s + 0.249·79-s + 0.756·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.543287205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.543287205\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 + 10.3T + 343T^{2} \) |
| 11 | \( 1 - 57.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 16.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 57.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 175.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 280.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 157.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 457.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 18.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 370.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 416.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 867.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 37.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 83.6T + 3.57e5T^{2} \) |
| 73 | \( 1 + 12.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 175.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 572.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 622.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.21e3T + 9.12e5T^{2} \) |
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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
−9.394756352392719004910094790711, −8.467228282088901245725677048908, −7.40572822470370450200737239686, −6.73060017982676221252337766656, −5.89073612469584847112310058556, −5.11527604403730804942307575895, −3.87034662897963979607725059497, −3.18621961135979935708094206377, −1.84756298653822527608429440287, −0.818345420903198442924507445162,
0.818345420903198442924507445162, 1.84756298653822527608429440287, 3.18621961135979935708094206377, 3.87034662897963979607725059497, 5.11527604403730804942307575895, 5.89073612469584847112310058556, 6.73060017982676221252337766656, 7.40572822470370450200737239686, 8.467228282088901245725677048908, 9.394756352392719004910094790711
