| L(s) = 1 | + 6.03·2-s + 3.45·3-s + 4.42·4-s − 25·5-s + 20.8·6-s + 221.·7-s − 166.·8-s − 231.·9-s − 150.·10-s − 121·11-s + 15.2·12-s + 547.·13-s + 1.33e3·14-s − 86.3·15-s − 1.14e3·16-s + 409.·17-s − 1.39e3·18-s + 361·19-s − 110.·20-s + 763.·21-s − 730.·22-s − 1.42e3·23-s − 575.·24-s + 625·25-s + 3.30e3·26-s − 1.63e3·27-s + 977.·28-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.221·3-s + 0.138·4-s − 0.447·5-s + 0.236·6-s + 1.70·7-s − 0.919·8-s − 0.950·9-s − 0.477·10-s − 0.301·11-s + 0.0306·12-s + 0.897·13-s + 1.81·14-s − 0.0991·15-s − 1.11·16-s + 0.343·17-s − 1.01·18-s + 0.229·19-s − 0.0617·20-s + 0.377·21-s − 0.321·22-s − 0.560·23-s − 0.203·24-s + 0.200·25-s + 0.957·26-s − 0.432·27-s + 0.235·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.906772576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.906772576\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 + 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 6.03T + 32T^{2} \) |
| 3 | \( 1 - 3.45T + 243T^{2} \) |
| 7 | \( 1 - 221.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 547.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 409.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 1.42e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 220.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.36e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.32e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.98e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.88e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.39e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.69e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.27e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.64e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.98e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.55e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.43e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.13e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.84e4T + 8.58e9T^{2} \) |
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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
−8.758873473444262237776706472953, −8.434753333471025729453517191393, −7.68098836342393024518303142826, −6.38003888813219696487626573689, −5.45335531072646994847788535151, −4.90014711570981142391199431514, −3.98516432934098683306552181216, −3.15699952600122778330271833939, −2.05135111196883692430260571632, −0.71933342693446642288893785261,
0.71933342693446642288893785261, 2.05135111196883692430260571632, 3.15699952600122778330271833939, 3.98516432934098683306552181216, 4.90014711570981142391199431514, 5.45335531072646994847788535151, 6.38003888813219696487626573689, 7.68098836342393024518303142826, 8.434753333471025729453517191393, 8.758873473444262237776706472953
