| L(s) = 1 | + 0.0692·3-s + 0.775·5-s + 4.19·7-s − 2.99·9-s + 0.210·11-s − 5.86·13-s + 0.0536·15-s − 6.81·17-s + 7.37·19-s + 0.290·21-s − 3.64·23-s − 4.39·25-s − 0.414·27-s − 6.37·29-s + 8.26·31-s + 0.0145·33-s + 3.25·35-s + 5.38·37-s − 0.405·39-s + 7.37·41-s + 4.77·43-s − 2.32·45-s + 7.67·47-s + 10.6·49-s − 0.471·51-s − 4.45·53-s + 0.163·55-s + ⋯ |
| L(s) = 1 | + 0.0399·3-s + 0.346·5-s + 1.58·7-s − 0.998·9-s + 0.0634·11-s − 1.62·13-s + 0.0138·15-s − 1.65·17-s + 1.69·19-s + 0.0634·21-s − 0.759·23-s − 0.879·25-s − 0.0798·27-s − 1.18·29-s + 1.48·31-s + 0.00253·33-s + 0.550·35-s + 0.884·37-s − 0.0650·39-s + 1.15·41-s + 0.728·43-s − 0.346·45-s + 1.11·47-s + 1.51·49-s − 0.0660·51-s − 0.611·53-s + 0.0220·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0692T + 3T^{2} \) |
| 5 | \( 1 - 0.775T + 5T^{2} \) |
| 7 | \( 1 - 4.19T + 7T^{2} \) |
| 11 | \( 1 - 0.210T + 11T^{2} \) |
| 13 | \( 1 + 5.86T + 13T^{2} \) |
| 17 | \( 1 + 6.81T + 17T^{2} \) |
| 19 | \( 1 - 7.37T + 19T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 + 6.37T + 29T^{2} \) |
| 31 | \( 1 - 8.26T + 31T^{2} \) |
| 37 | \( 1 - 5.38T + 37T^{2} \) |
| 41 | \( 1 - 7.37T + 41T^{2} \) |
| 43 | \( 1 - 4.77T + 43T^{2} \) |
| 47 | \( 1 - 7.67T + 47T^{2} \) |
| 53 | \( 1 + 4.45T + 53T^{2} \) |
| 59 | \( 1 + 3.83T + 59T^{2} \) |
| 61 | \( 1 - 9.08T + 61T^{2} \) |
| 67 | \( 1 + 15.5T + 67T^{2} \) |
| 71 | \( 1 + 8.49T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 3.41T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 9.76T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−7.69736041626386321914469772089, −6.94533174851322246362709974436, −5.76851698414793833859328260139, −5.55956627675613887051663993094, −4.57153032581519797271088127261, −4.23442238833383665936806759769, −2.72399251945479654530603745870, −2.37350946276831708838810183094, −1.38371228789165629969628414468, 0,
1.38371228789165629969628414468, 2.37350946276831708838810183094, 2.72399251945479654530603745870, 4.23442238833383665936806759769, 4.57153032581519797271088127261, 5.55956627675613887051663993094, 5.76851698414793833859328260139, 6.94533174851322246362709974436, 7.69736041626386321914469772089
