| L(s) = 1 | − 2-s − 3-s + 4-s − 2.71·5-s + 6-s − 2.34·7-s − 8-s + 9-s + 2.71·10-s + 0.805·11-s − 12-s − 13-s + 2.34·14-s + 2.71·15-s + 16-s − 7.05·17-s − 18-s + 3.05·19-s − 2.71·20-s + 2.34·21-s − 0.805·22-s − 3.54·23-s + 24-s + 2.36·25-s + 26-s − 27-s − 2.34·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.21·5-s + 0.408·6-s − 0.885·7-s − 0.353·8-s + 0.333·9-s + 0.858·10-s + 0.242·11-s − 0.288·12-s − 0.277·13-s + 0.625·14-s + 0.700·15-s + 0.250·16-s − 1.71·17-s − 0.235·18-s + 0.699·19-s − 0.606·20-s + 0.511·21-s − 0.171·22-s − 0.740·23-s + 0.204·24-s + 0.472·25-s + 0.196·26-s − 0.192·27-s − 0.442·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.03817289577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03817289577\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
| good | 5 | \( 1 + 2.71T + 5T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 11 | \( 1 - 0.805T + 11T^{2} \) |
| 17 | \( 1 + 7.05T + 17T^{2} \) |
| 19 | \( 1 - 3.05T + 19T^{2} \) |
| 23 | \( 1 + 3.54T + 23T^{2} \) |
| 29 | \( 1 + 1.98T + 29T^{2} \) |
| 31 | \( 1 + 7.06T + 31T^{2} \) |
| 37 | \( 1 - 1.93T + 37T^{2} \) |
| 41 | \( 1 + 1.08T + 41T^{2} \) |
| 43 | \( 1 + 3.41T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 - 6.23T + 53T^{2} \) |
| 59 | \( 1 - 1.79T + 59T^{2} \) |
| 61 | \( 1 + 2.31T + 61T^{2} \) |
| 67 | \( 1 - 4.26T + 67T^{2} \) |
| 71 | \( 1 + 0.749T + 71T^{2} \) |
| 73 | \( 1 + 4.06T + 73T^{2} \) |
| 79 | \( 1 - 4.75T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 7.49T + 89T^{2} \) |
| 97 | \( 1 + 9.03T + 97T^{2} \) |
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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
−7.85508006125008387349032926978, −7.01116063758136092380181126694, −6.76742229937300588907127651992, −5.93310780162110365023600779128, −5.05923498600809424759956253172, −4.13136918130837017469653634239, −3.60898281707651671114876764607, −2.63523033951027529707240244244, −1.54681861106467407428261788737, −0.10995244218046691468045906696,
0.10995244218046691468045906696, 1.54681861106467407428261788737, 2.63523033951027529707240244244, 3.60898281707651671114876764607, 4.13136918130837017469653634239, 5.05923498600809424759956253172, 5.93310780162110365023600779128, 6.76742229937300588907127651992, 7.01116063758136092380181126694, 7.85508006125008387349032926978
