| L(s) = 1 | + 1.11·2-s + 3·3-s − 6.74·4-s − 4.70·5-s + 3.35·6-s − 16.4·8-s + 9·9-s − 5.26·10-s − 11·11-s − 20.2·12-s + 71.0·13-s − 14.1·15-s + 35.5·16-s − 60.5·17-s + 10.0·18-s − 40.2·19-s + 31.7·20-s − 12.3·22-s − 86.1·23-s − 49.4·24-s − 102.·25-s + 79.4·26-s + 27·27-s − 0.964·29-s − 15.7·30-s + 85.8·31-s + 171.·32-s + ⋯ |
| L(s) = 1 | + 0.395·2-s + 0.577·3-s − 0.843·4-s − 0.420·5-s + 0.228·6-s − 0.729·8-s + 0.333·9-s − 0.166·10-s − 0.301·11-s − 0.487·12-s + 1.51·13-s − 0.242·15-s + 0.555·16-s − 0.864·17-s + 0.131·18-s − 0.486·19-s + 0.354·20-s − 0.119·22-s − 0.780·23-s − 0.420·24-s − 0.823·25-s + 0.599·26-s + 0.192·27-s − 0.00617·29-s − 0.0960·30-s + 0.497·31-s + 0.948·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.964601560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.964601560\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 - 1.11T + 8T^{2} \) |
| 5 | \( 1 + 4.70T + 125T^{2} \) |
| 13 | \( 1 - 71.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 60.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 40.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 86.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 0.964T + 2.43e4T^{2} \) |
| 31 | \( 1 - 85.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 8.62T + 5.06e4T^{2} \) |
| 41 | \( 1 - 278.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 322.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 76.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 344.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 57.4T + 2.05e5T^{2} \) |
| 61 | \( 1 + 224.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 591.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 698.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 58.6T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.03e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 108.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 402.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 780.T + 9.12e5T^{2} \) |
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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
−8.824570258057834848866581074723, −8.392112958638262416220987423208, −7.67745595191355484985127968652, −6.45478278813868945550398970439, −5.77961447983863577046870979748, −4.61218142964073670297878163353, −3.98171034014322168601303059053, −3.29603995760933012278476224476, −2.03172078870514483894019847069, −0.61476278572854979822024644882,
0.61476278572854979822024644882, 2.03172078870514483894019847069, 3.29603995760933012278476224476, 3.98171034014322168601303059053, 4.61218142964073670297878163353, 5.77961447983863577046870979748, 6.45478278813868945550398970439, 7.67745595191355484985127968652, 8.392112958638262416220987423208, 8.824570258057834848866581074723
