L(s) = 1 | − 1.64·2-s − 7.74·3-s − 5.30·4-s − 5·5-s + 12.7·6-s − 5.31·7-s + 21.8·8-s + 32.9·9-s + 8.21·10-s + 11·11-s + 41.0·12-s − 12.2·13-s + 8.72·14-s + 38.7·15-s + 6.55·16-s − 51.3·17-s − 54.1·18-s + 19·19-s + 26.5·20-s + 41.1·21-s − 18.0·22-s + 48.6·23-s − 169.·24-s + 25·25-s + 20.1·26-s − 46.3·27-s + 28.1·28-s + ⋯ |
L(s) = 1 | − 0.580·2-s − 1.49·3-s − 0.662·4-s − 0.447·5-s + 0.865·6-s − 0.286·7-s + 0.965·8-s + 1.22·9-s + 0.259·10-s + 0.301·11-s + 0.988·12-s − 0.262·13-s + 0.166·14-s + 0.666·15-s + 0.102·16-s − 0.731·17-s − 0.709·18-s + 0.229·19-s + 0.296·20-s + 0.427·21-s − 0.175·22-s + 0.441·23-s − 1.43·24-s + 0.200·25-s + 0.152·26-s − 0.330·27-s + 0.190·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.2289937975\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2289937975\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 - 19T \) |
good | 2 | \( 1 + 1.64T + 8T^{2} \) |
| 3 | \( 1 + 7.74T + 27T^{2} \) |
| 7 | \( 1 + 5.31T + 343T^{2} \) |
| 13 | \( 1 + 12.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 51.3T + 4.91e3T^{2} \) |
| 23 | \( 1 - 48.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 123.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 35.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 290.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 324.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 237.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 548.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 41.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 464.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 571.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 314.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 91.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 774.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.41e3T + 9.12e5T^{2} \) |
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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
−9.643039814581400249453514463369, −8.806274373096362659257412019492, −7.900259587914048474842111190762, −6.91937207631952787739142980099, −6.25068194290567748784115375711, −5.03072636525944610454589975741, −4.64846834235566075692888868616, −3.41249781906547417090645171527, −1.47667695710729194156581411802, −0.31924195466064402083718879371,
0.31924195466064402083718879371, 1.47667695710729194156581411802, 3.41249781906547417090645171527, 4.64846834235566075692888868616, 5.03072636525944610454589975741, 6.25068194290567748784115375711, 6.91937207631952787739142980099, 7.900259587914048474842111190762, 8.806274373096362659257412019492, 9.643039814581400249453514463369