| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 3.73·7-s + 8-s + 9-s + 10-s + 12-s + 3·13-s + 3.73·14-s + 15-s + 16-s + 0.267·17-s + 18-s − 4.46·19-s + 20-s + 3.73·21-s − 4.73·23-s + 24-s + 25-s + 3·26-s + 27-s + 3.73·28-s − 3·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.41·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.288·12-s + 0.832·13-s + 0.997·14-s + 0.258·15-s + 0.250·16-s + 0.0649·17-s + 0.235·18-s − 1.02·19-s + 0.223·20-s + 0.814·21-s − 0.986·23-s + 0.204·24-s + 0.200·25-s + 0.588·26-s + 0.192·27-s + 0.705·28-s − 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.019941541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.019941541\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 3.73T + 7T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 0.267T + 17T^{2} \) |
| 19 | \( 1 + 4.46T + 19T^{2} \) |
| 23 | \( 1 + 4.73T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 8.19T + 31T^{2} \) |
| 37 | \( 1 + 1.73T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 - 6.73T + 43T^{2} \) |
| 47 | \( 1 - 8.92T + 47T^{2} \) |
| 53 | \( 1 + 5.26T + 53T^{2} \) |
| 59 | \( 1 + 4.19T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 + 9.92T + 71T^{2} \) |
| 73 | \( 1 - 6.19T + 73T^{2} \) |
| 79 | \( 1 - 0.535T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 18.7T + 97T^{2} \) |
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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
−8.387457645867617326135688056787, −7.910554230765527998978303499899, −7.07747027785791675460445911131, −6.09156048186245265312117252397, −5.60901656647989014352336919055, −4.40148008582044716569074615489, −4.23729570019971350028836290756, −2.94969540625736663893497656088, −2.08239217263590813845521553259, −1.33019107023578876903192351829,
1.33019107023578876903192351829, 2.08239217263590813845521553259, 2.94969540625736663893497656088, 4.23729570019971350028836290756, 4.40148008582044716569074615489, 5.60901656647989014352336919055, 6.09156048186245265312117252397, 7.07747027785791675460445911131, 7.910554230765527998978303499899, 8.387457645867617326135688056787
