| L(s) = 1 | + 0.284·2-s − 0.0778·3-s − 1.91·4-s + 5-s − 0.0221·6-s + 1.40·7-s − 1.11·8-s − 2.99·9-s + 0.284·10-s − 1.61·11-s + 0.149·12-s + 2.02·13-s + 0.399·14-s − 0.0778·15-s + 3.52·16-s + 0.233·17-s − 0.852·18-s + 0.759·19-s − 1.91·20-s − 0.109·21-s − 0.459·22-s − 2.54·23-s + 0.0868·24-s + 25-s + 0.577·26-s + 0.466·27-s − 2.69·28-s + ⋯ |
| L(s) = 1 | + 0.201·2-s − 0.0449·3-s − 0.959·4-s + 0.447·5-s − 0.00904·6-s + 0.530·7-s − 0.394·8-s − 0.997·9-s + 0.0900·10-s − 0.486·11-s + 0.0431·12-s + 0.562·13-s + 0.106·14-s − 0.0201·15-s + 0.880·16-s + 0.0565·17-s − 0.200·18-s + 0.174·19-s − 0.429·20-s − 0.0238·21-s − 0.0979·22-s − 0.529·23-s + 0.0177·24-s + 0.200·25-s + 0.113·26-s + 0.0898·27-s − 0.509·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.487757144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.487757144\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 0.284T + 2T^{2} \) |
| 3 | \( 1 + 0.0778T + 3T^{2} \) |
| 7 | \( 1 - 1.40T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 - 2.02T + 13T^{2} \) |
| 17 | \( 1 - 0.233T + 17T^{2} \) |
| 19 | \( 1 - 0.759T + 19T^{2} \) |
| 23 | \( 1 + 2.54T + 23T^{2} \) |
| 29 | \( 1 - 3.33T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 + 2.89T + 37T^{2} \) |
| 41 | \( 1 - 2.33T + 41T^{2} \) |
| 43 | \( 1 + 0.527T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 - 5.10T + 59T^{2} \) |
| 61 | \( 1 - 8.78T + 61T^{2} \) |
| 67 | \( 1 - 3.79T + 67T^{2} \) |
| 71 | \( 1 - 5.56T + 71T^{2} \) |
| 73 | \( 1 - 9.80T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 - 5.15T + 89T^{2} \) |
| 97 | \( 1 + 4.54T + 97T^{2} \) |
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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
−9.044759129829721739523279219026, −8.398140694889738566384000711031, −7.912703336939521917379227938370, −6.62731879928545704337456497326, −5.69563606245775708011210824826, −5.26499867963526199525337650681, −4.32764082464945280185489457332, −3.35920561328539712619190743727, −2.31472383386454054382647569432, −0.799284454228382750506339283212,
0.799284454228382750506339283212, 2.31472383386454054382647569432, 3.35920561328539712619190743727, 4.32764082464945280185489457332, 5.26499867963526199525337650681, 5.69563606245775708011210824826, 6.62731879928545704337456497326, 7.912703336939521917379227938370, 8.398140694889738566384000711031, 9.044759129829721739523279219026
