L(s) = 1 | − 4.96·2-s + 3·3-s + 16.6·4-s + 13.1·5-s − 14.8·6-s − 5.77·7-s − 42.7·8-s + 9·9-s − 65.4·10-s + 25.4·11-s + 49.8·12-s − 88.8·13-s + 28.6·14-s + 39.5·15-s + 79.1·16-s + 87.3·17-s − 44.6·18-s + 142.·19-s + 219.·20-s − 17.3·21-s − 126.·22-s − 23·23-s − 128.·24-s + 49.2·25-s + 440.·26-s + 27·27-s − 95.9·28-s + ⋯ |
L(s) = 1 | − 1.75·2-s + 0.577·3-s + 2.07·4-s + 1.18·5-s − 1.01·6-s − 0.311·7-s − 1.88·8-s + 0.333·9-s − 2.07·10-s + 0.697·11-s + 1.19·12-s − 1.89·13-s + 0.546·14-s + 0.681·15-s + 1.23·16-s + 1.24·17-s − 0.584·18-s + 1.71·19-s + 2.45·20-s − 0.180·21-s − 1.22·22-s − 0.208·23-s − 1.09·24-s + 0.393·25-s + 3.32·26-s + 0.192·27-s − 0.647·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.544009920\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.544009920\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 + 23T \) |
| 29 | \( 1 - 29T \) |
good | 2 | \( 1 + 4.96T + 8T^{2} \) |
| 5 | \( 1 - 13.1T + 125T^{2} \) |
| 7 | \( 1 + 5.77T + 343T^{2} \) |
| 11 | \( 1 - 25.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 88.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 87.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 142.T + 6.85e3T^{2} \) |
| 31 | \( 1 + 69.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 337.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 345.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 35.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 138.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 406.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 128.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 10.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 709.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 176.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 546.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 689.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 70.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.51e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 917.T + 9.12e5T^{2} \) |
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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
−9.236107562960431474704411921285, −8.016910519927972108824911781171, −7.50020888723764482633832732120, −6.83190931817589908831719743291, −5.90447058985596385250357830405, −4.97264069546362392800296484748, −3.30930156508571826047068562350, −2.47559691207077438463987411128, −1.66483958443561076213765884210, −0.74590027478178405279297838185,
0.74590027478178405279297838185, 1.66483958443561076213765884210, 2.47559691207077438463987411128, 3.30930156508571826047068562350, 4.97264069546362392800296484748, 5.90447058985596385250357830405, 6.83190931817589908831719743291, 7.50020888723764482633832732120, 8.016910519927972108824911781171, 9.236107562960431474704411921285