| L(s) = 1 | + 2-s + 2.65·3-s + 4-s − 5-s + 2.65·6-s − 2.26·7-s + 8-s + 4.03·9-s − 10-s + 11-s + 2.65·12-s − 0.447·13-s − 2.26·14-s − 2.65·15-s + 16-s − 0.343·17-s + 4.03·18-s − 1.43·19-s − 20-s − 6.00·21-s + 22-s + 3.74·23-s + 2.65·24-s + 25-s − 0.447·26-s + 2.75·27-s − 2.26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.53·3-s + 0.5·4-s − 0.447·5-s + 1.08·6-s − 0.855·7-s + 0.353·8-s + 1.34·9-s − 0.316·10-s + 0.301·11-s + 0.765·12-s − 0.124·13-s − 0.604·14-s − 0.685·15-s + 0.250·16-s − 0.0832·17-s + 0.951·18-s − 0.328·19-s − 0.223·20-s − 1.30·21-s + 0.213·22-s + 0.781·23-s + 0.541·24-s + 0.200·25-s − 0.0878·26-s + 0.530·27-s − 0.427·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.203455470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.203455470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 - 2.65T + 3T^{2} \) |
| 7 | \( 1 + 2.26T + 7T^{2} \) |
| 13 | \( 1 + 0.447T + 13T^{2} \) |
| 17 | \( 1 + 0.343T + 17T^{2} \) |
| 19 | \( 1 + 1.43T + 19T^{2} \) |
| 23 | \( 1 - 3.74T + 23T^{2} \) |
| 29 | \( 1 - 0.198T + 29T^{2} \) |
| 31 | \( 1 - 7.43T + 31T^{2} \) |
| 37 | \( 1 - 7.75T + 37T^{2} \) |
| 41 | \( 1 - 7.03T + 41T^{2} \) |
| 43 | \( 1 - 4.16T + 43T^{2} \) |
| 47 | \( 1 - 1.99T + 47T^{2} \) |
| 53 | \( 1 + 1.02T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 3.10T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 4.10T + 71T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 1.76T + 83T^{2} \) |
| 89 | \( 1 - 0.676T + 89T^{2} \) |
| 97 | \( 1 - 0.741T + 97T^{2} \) |
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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
−7.74704483769887194471093588965, −7.24231491934039752206868927893, −6.51055538494284187919635922907, −5.85292228461742470334863721519, −4.67768816802511546027401151437, −4.14015368272534117170910782615, −3.42935741861621841098217905178, −2.82198141258200981902889936792, −2.26139941938822737903974244904, −0.949359911126973668814889258392,
0.949359911126973668814889258392, 2.26139941938822737903974244904, 2.82198141258200981902889936792, 3.42935741861621841098217905178, 4.14015368272534117170910782615, 4.67768816802511546027401151437, 5.85292228461742470334863721519, 6.51055538494284187919635922907, 7.24231491934039752206868927893, 7.74704483769887194471093588965
