L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 0.449·7-s + 8-s + 9-s − 1.44·11-s + 12-s + 2.44·13-s − 0.449·14-s + 16-s − 0.449·17-s + 18-s + 19-s − 0.449·21-s − 1.44·22-s − 23-s + 24-s + 2.44·26-s + 27-s − 0.449·28-s + 10.3·29-s − 3·31-s + 32-s − 1.44·33-s − 0.449·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s − 0.169·7-s + 0.353·8-s + 0.333·9-s − 0.437·11-s + 0.288·12-s + 0.679·13-s − 0.120·14-s + 0.250·16-s − 0.109·17-s + 0.235·18-s + 0.229·19-s − 0.0980·21-s − 0.309·22-s − 0.208·23-s + 0.204·24-s + 0.480·26-s + 0.192·27-s − 0.0849·28-s + 1.92·29-s − 0.538·31-s + 0.176·32-s − 0.252·33-s − 0.0770·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.735453367\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.735453367\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 0.449T + 7T^{2} \) |
| 11 | \( 1 + 1.44T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 + 0.449T + 17T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 - 10.3T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 8.89T + 41T^{2} \) |
| 43 | \( 1 - 2.44T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 2.55T + 53T^{2} \) |
| 59 | \( 1 - 1.55T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 + 9.24T + 67T^{2} \) |
| 71 | \( 1 + 6.44T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 5T + 79T^{2} \) |
| 83 | \( 1 + 6.34T + 83T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 - 6.44T + 97T^{2} \) |
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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
−8.737758727543438572287638875497, −7.911963501914263737797331977426, −7.34322251645105806821781545156, −6.32428577261191538069835441484, −5.81313881437308006709748361570, −4.68096876548028483421755905033, −4.08927976574825953133204088878, −3.06105256458612005816213438089, −2.44451831551702880233888861245, −1.11114235008707996922985062677,
1.11114235008707996922985062677, 2.44451831551702880233888861245, 3.06105256458612005816213438089, 4.08927976574825953133204088878, 4.68096876548028483421755905033, 5.81313881437308006709748361570, 6.32428577261191538069835441484, 7.34322251645105806821781545156, 7.911963501914263737797331977426, 8.737758727543438572287638875497