| L(s) = 1 | − 2.40·2-s − 2.20·4-s + 19.1·5-s + 10.9·7-s + 24.5·8-s − 46.2·10-s + 60.8·11-s + 37.3·13-s − 26.4·14-s − 41.5·16-s − 84.1·17-s − 95.5·19-s − 42.2·20-s − 146.·22-s − 27.4·23-s + 243.·25-s − 89.8·26-s − 24.1·28-s + 38.3·29-s + 16.7·31-s − 96.5·32-s + 202.·34-s + 210.·35-s + 125.·37-s + 230.·38-s + 471.·40-s + 336.·41-s + ⋯ |
| L(s) = 1 | − 0.851·2-s − 0.275·4-s + 1.71·5-s + 0.592·7-s + 1.08·8-s − 1.46·10-s + 1.66·11-s + 0.796·13-s − 0.504·14-s − 0.648·16-s − 1.20·17-s − 1.15·19-s − 0.472·20-s − 1.41·22-s − 0.248·23-s + 1.94·25-s − 0.677·26-s − 0.163·28-s + 0.245·29-s + 0.0972·31-s − 0.533·32-s + 1.02·34-s + 1.01·35-s + 0.556·37-s + 0.982·38-s + 1.86·40-s + 1.28·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.366832646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.366832646\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 2.40T + 8T^{2} \) |
| 5 | \( 1 - 19.1T + 125T^{2} \) |
| 7 | \( 1 - 10.9T + 343T^{2} \) |
| 11 | \( 1 - 60.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 95.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 27.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 38.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 16.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 125.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 336.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 272.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 344.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 441.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 509.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 77.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 369.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 358.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.12e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 814.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 69.8T + 7.04e5T^{2} \) |
| 97 | \( 1 - 623.T + 9.12e5T^{2} \) |
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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
−8.905966247343545374268709466412, −8.367145173886362474703384059591, −7.14299563863869002461422238103, −6.33710114957596666602888114291, −5.83144342177330741749648937340, −4.59180854653420922028861260844, −4.06229541482991588912062099099, −2.33858704148244527705150365006, −1.62520807558826106426829788599, −0.877305691080237974147493052717,
0.877305691080237974147493052717, 1.62520807558826106426829788599, 2.33858704148244527705150365006, 4.06229541482991588912062099099, 4.59180854653420922028861260844, 5.83144342177330741749648937340, 6.33710114957596666602888114291, 7.14299563863869002461422238103, 8.367145173886362474703384059591, 8.905966247343545374268709466412
