| L(s) = 1 | − 3-s + 2.28·5-s − 2.96·7-s + 9-s + 4.55·11-s + 0.515·13-s − 2.28·15-s − 7.19·17-s + 0.0161·19-s + 2.96·21-s + 3.87·23-s + 0.214·25-s − 27-s − 0.454·29-s − 7.42·31-s − 4.55·33-s − 6.77·35-s + 4.57·37-s − 0.515·39-s − 3.91·41-s + 4.25·43-s + 2.28·45-s + 2.95·47-s + 1.81·49-s + 7.19·51-s − 9.31·53-s + 10.3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.02·5-s − 1.12·7-s + 0.333·9-s + 1.37·11-s + 0.143·13-s − 0.589·15-s − 1.74·17-s + 0.00369·19-s + 0.647·21-s + 0.808·23-s + 0.0429·25-s − 0.192·27-s − 0.0843·29-s − 1.33·31-s − 0.792·33-s − 1.14·35-s + 0.752·37-s − 0.0826·39-s − 0.612·41-s + 0.648·43-s + 0.340·45-s + 0.430·47-s + 0.258·49-s + 1.00·51-s − 1.27·53-s + 1.40·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 - T \) |
| good | 5 | \( 1 - 2.28T + 5T^{2} \) |
| 7 | \( 1 + 2.96T + 7T^{2} \) |
| 11 | \( 1 - 4.55T + 11T^{2} \) |
| 13 | \( 1 - 0.515T + 13T^{2} \) |
| 17 | \( 1 + 7.19T + 17T^{2} \) |
| 19 | \( 1 - 0.0161T + 19T^{2} \) |
| 23 | \( 1 - 3.87T + 23T^{2} \) |
| 29 | \( 1 + 0.454T + 29T^{2} \) |
| 31 | \( 1 + 7.42T + 31T^{2} \) |
| 37 | \( 1 - 4.57T + 37T^{2} \) |
| 41 | \( 1 + 3.91T + 41T^{2} \) |
| 43 | \( 1 - 4.25T + 43T^{2} \) |
| 47 | \( 1 - 2.95T + 47T^{2} \) |
| 53 | \( 1 + 9.31T + 53T^{2} \) |
| 59 | \( 1 + 5.55T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 5.15T + 67T^{2} \) |
| 71 | \( 1 + 2.60T + 71T^{2} \) |
| 73 | \( 1 - 7.70T + 73T^{2} \) |
| 79 | \( 1 + 6.85T + 79T^{2} \) |
| 83 | \( 1 + 6.44T + 83T^{2} \) |
| 89 | \( 1 + 2.50T + 89T^{2} \) |
| 97 | \( 1 - 8.63T + 97T^{2} \) |
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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
−7.44938267737153587939479931668, −6.72941920292335832195693312632, −6.30250633701553107502178088292, −5.86180688668067377533042529701, −4.85640222732761457586571126024, −4.09748082979522850246299929596, −3.25063347401779560908883842238, −2.20266996441174951209809457193, −1.34695053114204813432586721143, 0,
1.34695053114204813432586721143, 2.20266996441174951209809457193, 3.25063347401779560908883842238, 4.09748082979522850246299929596, 4.85640222732761457586571126024, 5.86180688668067377533042529701, 6.30250633701553107502178088292, 6.72941920292335832195693312632, 7.44938267737153587939479931668
