L(s) = 1 | + 0.508·2-s − 0.408·3-s − 1.74·4-s + 4.07·5-s − 0.207·6-s + 0.983·7-s − 1.90·8-s − 2.83·9-s + 2.07·10-s + 6.22·11-s + 0.711·12-s − 13-s + 0.499·14-s − 1.66·15-s + 2.51·16-s + 0.405·17-s − 1.44·18-s − 5.40·19-s − 7.09·20-s − 0.401·21-s + 3.16·22-s + 5.24·23-s + 0.776·24-s + 11.6·25-s − 0.508·26-s + 2.38·27-s − 1.71·28-s + ⋯ |
L(s) = 1 | + 0.359·2-s − 0.235·3-s − 0.870·4-s + 1.82·5-s − 0.0847·6-s + 0.371·7-s − 0.672·8-s − 0.944·9-s + 0.655·10-s + 1.87·11-s + 0.205·12-s − 0.277·13-s + 0.133·14-s − 0.429·15-s + 0.629·16-s + 0.0983·17-s − 0.339·18-s − 1.24·19-s − 1.58·20-s − 0.0876·21-s + 0.674·22-s + 1.09·23-s + 0.158·24-s + 2.32·25-s − 0.0996·26-s + 0.458·27-s − 0.323·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.175267514\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.175267514\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 0.508T + 2T^{2} \) |
| 3 | \( 1 + 0.408T + 3T^{2} \) |
| 5 | \( 1 - 4.07T + 5T^{2} \) |
| 7 | \( 1 - 0.983T + 7T^{2} \) |
| 11 | \( 1 - 6.22T + 11T^{2} \) |
| 17 | \( 1 - 0.405T + 17T^{2} \) |
| 19 | \( 1 + 5.40T + 19T^{2} \) |
| 23 | \( 1 - 5.24T + 23T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 - 7.38T + 31T^{2} \) |
| 37 | \( 1 - 1.40T + 37T^{2} \) |
| 41 | \( 1 + 2.27T + 41T^{2} \) |
| 43 | \( 1 + 8.17T + 43T^{2} \) |
| 47 | \( 1 + 7.54T + 47T^{2} \) |
| 53 | \( 1 - 6.06T + 53T^{2} \) |
| 59 | \( 1 - 8.47T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 14.9T + 67T^{2} \) |
| 71 | \( 1 + 5.18T + 71T^{2} \) |
| 73 | \( 1 - 2.21T + 73T^{2} \) |
| 79 | \( 1 + 7.96T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 - 9.20T + 89T^{2} \) |
| 97 | \( 1 - 9.79T + 97T^{2} \) |
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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
−9.626095641719823737636214591490, −8.799536030861286445171716495944, −8.482771919918397809648804428192, −6.57058123728432497004173887967, −6.36423170356165137356740631928, −5.33114213992081929248943245073, −4.79154545368085819748265234669, −3.58602222781093707614229715506, −2.35968488987581986529713758794, −1.13087703003412826210454677670,
1.13087703003412826210454677670, 2.35968488987581986529713758794, 3.58602222781093707614229715506, 4.79154545368085819748265234669, 5.33114213992081929248943245073, 6.36423170356165137356740631928, 6.57058123728432497004173887967, 8.482771919918397809648804428192, 8.799536030861286445171716495944, 9.626095641719823737636214591490