| L(s) = 1 | + 2-s + 4-s − 4.24·7-s + 8-s − 1.42·11-s + 6.91·13-s − 4.24·14-s + 16-s − 5.10·17-s + 19-s − 1.42·22-s + 3.67·23-s + 6.91·26-s − 4.24·28-s − 8.10·29-s − 1.28·31-s + 32-s − 5.10·34-s + 0.856·37-s + 38-s + 8.01·41-s − 3.57·43-s − 1.42·44-s + 3.67·46-s − 3.81·47-s + 11.0·49-s + 6.91·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.60·7-s + 0.353·8-s − 0.430·11-s + 1.91·13-s − 1.13·14-s + 0.250·16-s − 1.23·17-s + 0.229·19-s − 0.304·22-s + 0.765·23-s + 1.35·26-s − 0.802·28-s − 1.50·29-s − 0.231·31-s + 0.176·32-s − 0.874·34-s + 0.140·37-s + 0.162·38-s + 1.25·41-s − 0.544·43-s − 0.215·44-s + 0.541·46-s − 0.556·47-s + 1.57·49-s + 0.959·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 + 1.42T + 11T^{2} \) |
| 13 | \( 1 - 6.91T + 13T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 23 | \( 1 - 3.67T + 23T^{2} \) |
| 29 | \( 1 + 8.10T + 29T^{2} \) |
| 31 | \( 1 + 1.28T + 31T^{2} \) |
| 37 | \( 1 - 0.856T + 37T^{2} \) |
| 41 | \( 1 - 8.01T + 41T^{2} \) |
| 43 | \( 1 + 3.57T + 43T^{2} \) |
| 47 | \( 1 + 3.81T + 47T^{2} \) |
| 53 | \( 1 - 9.06T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 8.20T + 61T^{2} \) |
| 67 | \( 1 + 4.38T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 5.38T + 73T^{2} \) |
| 79 | \( 1 - 2.14T + 79T^{2} \) |
| 83 | \( 1 - 1.04T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 6.81T + 97T^{2} \) |
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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
−7.10055176916182136889016000132, −6.71069816325063153335758091446, −5.90896633249390032780526893519, −5.65165485098506055893059094066, −4.46032401552084108776135170454, −3.79774528044929844846937749985, −3.22522014947094594637093432025, −2.49355133254412103781942907692, −1.33658403437998747671368320557, 0,
1.33658403437998747671368320557, 2.49355133254412103781942907692, 3.22522014947094594637093432025, 3.79774528044929844846937749985, 4.46032401552084108776135170454, 5.65165485098506055893059094066, 5.90896633249390032780526893519, 6.71069816325063153335758091446, 7.10055176916182136889016000132
