| L(s) = 1 | − 2.31·2-s + 1.57·3-s + 3.35·4-s − 1.84·5-s − 3.64·6-s + 3.43·7-s − 3.14·8-s − 0.525·9-s + 4.26·10-s + 0.201·11-s + 5.28·12-s − 13-s − 7.96·14-s − 2.89·15-s + 0.565·16-s − 2.22·17-s + 1.21·18-s − 5.08·19-s − 6.19·20-s + 5.41·21-s − 0.465·22-s − 1.10·23-s − 4.94·24-s − 1.60·25-s + 2.31·26-s − 5.54·27-s + 11.5·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 0.908·3-s + 1.67·4-s − 0.824·5-s − 1.48·6-s + 1.30·7-s − 1.11·8-s − 0.175·9-s + 1.34·10-s + 0.0606·11-s + 1.52·12-s − 0.277·13-s − 2.12·14-s − 0.748·15-s + 0.141·16-s − 0.540·17-s + 0.286·18-s − 1.16·19-s − 1.38·20-s + 1.18·21-s − 0.0993·22-s − 0.230·23-s − 1.01·24-s − 0.320·25-s + 0.454·26-s − 1.06·27-s + 2.18·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.31T + 2T^{2} \) |
| 3 | \( 1 - 1.57T + 3T^{2} \) |
| 5 | \( 1 + 1.84T + 5T^{2} \) |
| 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 - 0.201T + 11T^{2} \) |
| 17 | \( 1 + 2.22T + 17T^{2} \) |
| 19 | \( 1 + 5.08T + 19T^{2} \) |
| 23 | \( 1 + 1.10T + 23T^{2} \) |
| 29 | \( 1 + 9.16T + 29T^{2} \) |
| 31 | \( 1 - 9.57T + 31T^{2} \) |
| 37 | \( 1 + 5.38T + 37T^{2} \) |
| 41 | \( 1 - 6.22T + 41T^{2} \) |
| 43 | \( 1 + 1.75T + 43T^{2} \) |
| 47 | \( 1 + 7.84T + 47T^{2} \) |
| 53 | \( 1 + 6.31T + 53T^{2} \) |
| 59 | \( 1 + 8.81T + 59T^{2} \) |
| 61 | \( 1 - 5.31T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 + 2.37T + 71T^{2} \) |
| 73 | \( 1 + 5.85T + 73T^{2} \) |
| 79 | \( 1 - 8.04T + 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−8.982617837504011300079396042853, −8.310603233148550181767612486460, −7.999823732127662147827851017157, −7.36797492842532028023422136335, −6.29500293286672433973624458803, −4.82148895970918363993354416875, −3.84729207035120540304719164242, −2.46664359364733477813366745339, −1.69070790389590428959647784492, 0,
1.69070790389590428959647784492, 2.46664359364733477813366745339, 3.84729207035120540304719164242, 4.82148895970918363993354416875, 6.29500293286672433973624458803, 7.36797492842532028023422136335, 7.999823732127662147827851017157, 8.310603233148550181767612486460, 8.982617837504011300079396042853
