| L(s) = 1 | − 2-s − 1.33·3-s + 4-s + 2.98·5-s + 1.33·6-s + 1.91·7-s − 8-s − 1.22·9-s − 2.98·10-s + 2.33·11-s − 1.33·12-s − 1.24·13-s − 1.91·14-s − 3.98·15-s + 16-s + 4.75·17-s + 1.22·18-s − 6.69·19-s + 2.98·20-s − 2.55·21-s − 2.33·22-s + 2.52·23-s + 1.33·24-s + 3.93·25-s + 1.24·26-s + 5.62·27-s + 1.91·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.769·3-s + 0.5·4-s + 1.33·5-s + 0.544·6-s + 0.724·7-s − 0.353·8-s − 0.407·9-s − 0.945·10-s + 0.703·11-s − 0.384·12-s − 0.344·13-s − 0.512·14-s − 1.02·15-s + 0.250·16-s + 1.15·17-s + 0.288·18-s − 1.53·19-s + 0.668·20-s − 0.557·21-s − 0.497·22-s + 0.526·23-s + 0.272·24-s + 0.786·25-s + 0.243·26-s + 1.08·27-s + 0.362·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.432505158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.432505158\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 + 1.33T + 3T^{2} \) |
| 5 | \( 1 - 2.98T + 5T^{2} \) |
| 7 | \( 1 - 1.91T + 7T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 13 | \( 1 + 1.24T + 13T^{2} \) |
| 17 | \( 1 - 4.75T + 17T^{2} \) |
| 19 | \( 1 + 6.69T + 19T^{2} \) |
| 23 | \( 1 - 2.52T + 23T^{2} \) |
| 29 | \( 1 + 4.11T + 29T^{2} \) |
| 31 | \( 1 - 0.469T + 31T^{2} \) |
| 37 | \( 1 + 4.62T + 37T^{2} \) |
| 41 | \( 1 - 2.98T + 41T^{2} \) |
| 43 | \( 1 - 2.32T + 43T^{2} \) |
| 47 | \( 1 - 4.98T + 47T^{2} \) |
| 53 | \( 1 - 9.38T + 53T^{2} \) |
| 59 | \( 1 - 8.98T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 8.59T + 71T^{2} \) |
| 73 | \( 1 - 1.10T + 73T^{2} \) |
| 79 | \( 1 + 4.62T + 79T^{2} \) |
| 83 | \( 1 + 17.7T + 83T^{2} \) |
| 89 | \( 1 - 8.16T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−8.693381126161395506709884775593, −7.74343815944394552418005441439, −6.85079045187316616181105876604, −6.22856796727438968610250590220, −5.56979443023134519615075653281, −5.05990597739076372570582113340, −3.85619849387756190095225394618, −2.54029040648961335369282804241, −1.81252492117881077110817460763, −0.813933644174207759355812933856,
0.813933644174207759355812933856, 1.81252492117881077110817460763, 2.54029040648961335369282804241, 3.85619849387756190095225394618, 5.05990597739076372570582113340, 5.56979443023134519615075653281, 6.22856796727438968610250590220, 6.85079045187316616181105876604, 7.74343815944394552418005441439, 8.693381126161395506709884775593
