| L(s) = 1 | − 0.823·3-s + 3.63·5-s + 3.42·7-s − 2.32·9-s − 1.66·11-s + 1.71·13-s − 2.99·15-s − 17-s − 8.21·19-s − 2.81·21-s + 1.40·23-s + 8.24·25-s + 4.38·27-s − 9.95·29-s − 6.20·31-s + 1.37·33-s + 12.4·35-s − 1.91·37-s − 1.41·39-s + 4.14·41-s − 7.78·43-s − 8.45·45-s + 3.59·47-s + 4.71·49-s + 0.823·51-s − 9.21·53-s − 6.07·55-s + ⋯ |
| L(s) = 1 | − 0.475·3-s + 1.62·5-s + 1.29·7-s − 0.773·9-s − 0.503·11-s + 0.475·13-s − 0.773·15-s − 0.242·17-s − 1.88·19-s − 0.615·21-s + 0.293·23-s + 1.64·25-s + 0.843·27-s − 1.84·29-s − 1.11·31-s + 0.239·33-s + 2.10·35-s − 0.314·37-s − 0.226·39-s + 0.647·41-s − 1.18·43-s − 1.25·45-s + 0.524·47-s + 0.673·49-s + 0.115·51-s − 1.26·53-s − 0.819·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 0.823T + 3T^{2} \) |
| 5 | \( 1 - 3.63T + 5T^{2} \) |
| 7 | \( 1 - 3.42T + 7T^{2} \) |
| 11 | \( 1 + 1.66T + 11T^{2} \) |
| 13 | \( 1 - 1.71T + 13T^{2} \) |
| 19 | \( 1 + 8.21T + 19T^{2} \) |
| 23 | \( 1 - 1.40T + 23T^{2} \) |
| 29 | \( 1 + 9.95T + 29T^{2} \) |
| 31 | \( 1 + 6.20T + 31T^{2} \) |
| 37 | \( 1 + 1.91T + 37T^{2} \) |
| 41 | \( 1 - 4.14T + 41T^{2} \) |
| 43 | \( 1 + 7.78T + 43T^{2} \) |
| 47 | \( 1 - 3.59T + 47T^{2} \) |
| 53 | \( 1 + 9.21T + 53T^{2} \) |
| 61 | \( 1 + 9.69T + 61T^{2} \) |
| 67 | \( 1 + 7.29T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 - 4.25T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 + 7.87T + 83T^{2} \) |
| 89 | \( 1 - 6.32T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.53590213531335528033996951764, −6.48053143678569752814727548551, −6.07412691159735867152243605382, −5.36843939615508367711621976820, −5.00495747092808943081499864316, −4.06753378238644252868546630260, −2.86176501536207599151624196885, −1.97043571403121052710827279475, −1.59554641514916828341744469039, 0,
1.59554641514916828341744469039, 1.97043571403121052710827279475, 2.86176501536207599151624196885, 4.06753378238644252868546630260, 5.00495747092808943081499864316, 5.36843939615508367711621976820, 6.07412691159735867152243605382, 6.48053143678569752814727548551, 7.53590213531335528033996951764
