| L(s) = 1 | − 2.63·2-s − 2.81·3-s + 4.92·4-s + 1.41·5-s + 7.39·6-s + 3.18·7-s − 7.71·8-s + 4.89·9-s − 3.71·10-s + 0.715·11-s − 13.8·12-s − 1.65·13-s − 8.37·14-s − 3.96·15-s + 10.4·16-s + 6.51·17-s − 12.8·18-s + 19-s + 6.96·20-s − 8.93·21-s − 1.88·22-s − 3.69·23-s + 21.6·24-s − 3.00·25-s + 4.36·26-s − 5.33·27-s + 15.6·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 1.62·3-s + 2.46·4-s + 0.631·5-s + 3.02·6-s + 1.20·7-s − 2.72·8-s + 1.63·9-s − 1.17·10-s + 0.215·11-s − 3.99·12-s − 0.460·13-s − 2.23·14-s − 1.02·15-s + 2.61·16-s + 1.57·17-s − 3.03·18-s + 0.229·19-s + 1.55·20-s − 1.95·21-s − 0.401·22-s − 0.771·23-s + 4.42·24-s − 0.600·25-s + 0.856·26-s − 1.02·27-s + 2.96·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 - T \) |
| 317 | \( 1 - T \) |
| good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 3 | \( 1 + 2.81T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 - 0.715T + 11T^{2} \) |
| 13 | \( 1 + 1.65T + 13T^{2} \) |
| 17 | \( 1 - 6.51T + 17T^{2} \) |
| 23 | \( 1 + 3.69T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 - 3.26T + 31T^{2} \) |
| 37 | \( 1 + 9.57T + 37T^{2} \) |
| 41 | \( 1 + 0.712T + 41T^{2} \) |
| 43 | \( 1 - 7.35T + 43T^{2} \) |
| 47 | \( 1 + 6.54T + 47T^{2} \) |
| 53 | \( 1 - 11.8T + 53T^{2} \) |
| 59 | \( 1 - 9.77T + 59T^{2} \) |
| 61 | \( 1 + 6.93T + 61T^{2} \) |
| 67 | \( 1 + 8.41T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 6.26T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 - 7.34T + 89T^{2} \) |
| 97 | \( 1 + 18.8T + 97T^{2} \) |
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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
−7.77829196666278670891099085525, −7.13953832401210114302329985511, −6.46831480639408555548237388952, −5.62807351214179505957358689724, −5.37117845538223409926763839303, −4.22169831462026031909406368698, −2.71160623322451121348039043990, −1.56993161415032418415228135726, −1.21754778006697597639690432647, 0,
1.21754778006697597639690432647, 1.56993161415032418415228135726, 2.71160623322451121348039043990, 4.22169831462026031909406368698, 5.37117845538223409926763839303, 5.62807351214179505957358689724, 6.46831480639408555548237388952, 7.13953832401210114302329985511, 7.77829196666278670891099085525
