| L(s) = 1 | + 4.53·2-s + 3·3-s + 12.5·4-s + 5·5-s + 13.5·6-s − 7·7-s + 20.5·8-s + 9·9-s + 22.6·10-s − 19.0·11-s + 37.5·12-s − 2.93·13-s − 31.7·14-s + 15·15-s − 7.21·16-s − 6.49·17-s + 40.7·18-s − 5.43·19-s + 62.6·20-s − 21·21-s − 86.3·22-s + 49.3·23-s + 61.5·24-s + 25·25-s − 13.3·26-s + 27·27-s − 87.7·28-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 0.577·3-s + 1.56·4-s + 0.447·5-s + 0.924·6-s − 0.377·7-s + 0.907·8-s + 0.333·9-s + 0.716·10-s − 0.522·11-s + 0.904·12-s − 0.0626·13-s − 0.605·14-s + 0.258·15-s − 0.112·16-s − 0.0927·17-s + 0.533·18-s − 0.0656·19-s + 0.700·20-s − 0.218·21-s − 0.837·22-s + 0.447·23-s + 0.523·24-s + 0.200·25-s − 0.100·26-s + 0.192·27-s − 0.592·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.178010652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.178010652\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 4.53T + 8T^{2} \) |
| 11 | \( 1 + 19.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.93T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.49T + 4.91e3T^{2} \) |
| 19 | \( 1 + 5.43T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 291.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 244.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 193.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 315.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 300.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 86.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 509.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 83.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 5.25T + 2.26e5T^{2} \) |
| 67 | \( 1 - 205.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.00e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 863.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.33e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 326.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.52e3T + 9.12e5T^{2} \) |
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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
−13.30346175272847329227780832686, −12.73831602913954178485104586965, −11.53088546756880426510015652679, −10.24689496600514792997960725745, −8.959543556742975218076338319296, −7.35197583697436006371302737442, −6.13911297908769492799448079281, −4.97877979516232912120258346709, −3.60040878612527266076080746053, −2.36836063550997595589839875669,
2.36836063550997595589839875669, 3.60040878612527266076080746053, 4.97877979516232912120258346709, 6.13911297908769492799448079281, 7.35197583697436006371302737442, 8.959543556742975218076338319296, 10.24689496600514792997960725745, 11.53088546756880426510015652679, 12.73831602913954178485104586965, 13.30346175272847329227780832686
