| L(s) = 1 | − 1.67·2-s − 3-s + 0.810·4-s − 3.83·5-s + 1.67·6-s − 4.72·7-s + 1.99·8-s + 9-s + 6.43·10-s + 6.37·11-s − 0.810·12-s + 6.95·13-s + 7.92·14-s + 3.83·15-s − 4.96·16-s − 17-s − 1.67·18-s + 6.78·19-s − 3.11·20-s + 4.72·21-s − 10.6·22-s + 5.35·23-s − 1.99·24-s + 9.74·25-s − 11.6·26-s − 27-s − 3.83·28-s + ⋯ |
| L(s) = 1 | − 1.18·2-s − 0.577·3-s + 0.405·4-s − 1.71·5-s + 0.684·6-s − 1.78·7-s + 0.704·8-s + 0.333·9-s + 2.03·10-s + 1.92·11-s − 0.234·12-s + 1.92·13-s + 2.11·14-s + 0.991·15-s − 1.24·16-s − 0.242·17-s − 0.395·18-s + 1.55·19-s − 0.695·20-s + 1.03·21-s − 2.27·22-s + 1.11·23-s − 0.407·24-s + 1.94·25-s − 2.28·26-s − 0.192·27-s − 0.724·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5424873365\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5424873365\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 1.67T + 2T^{2} \) |
| 5 | \( 1 + 3.83T + 5T^{2} \) |
| 7 | \( 1 + 4.72T + 7T^{2} \) |
| 11 | \( 1 - 6.37T + 11T^{2} \) |
| 13 | \( 1 - 6.95T + 13T^{2} \) |
| 19 | \( 1 - 6.78T + 19T^{2} \) |
| 23 | \( 1 - 5.35T + 23T^{2} \) |
| 29 | \( 1 + 3.88T + 29T^{2} \) |
| 31 | \( 1 + 4.37T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 - 6.61T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 3.03T + 47T^{2} \) |
| 53 | \( 1 + 3.69T + 53T^{2} \) |
| 59 | \( 1 + 0.276T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 + 4.08T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 1.10T + 73T^{2} \) |
| 83 | \( 1 + 3.30T + 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.791493033730995340283055467893, −7.57844155987714965275401950133, −7.17037637111665214144050306240, −6.52261902214139315657403288052, −5.78444301915067707139070564270, −4.32930116863224973723948913070, −3.75919765903948967609821596808, −3.28509198689480904048877010075, −1.17753141718733857387006867298, −0.65315778291841160113057828433,
0.65315778291841160113057828433, 1.17753141718733857387006867298, 3.28509198689480904048877010075, 3.75919765903948967609821596808, 4.32930116863224973723948913070, 5.78444301915067707139070564270, 6.52261902214139315657403288052, 7.17037637111665214144050306240, 7.57844155987714965275401950133, 8.791493033730995340283055467893
