| L(s) = 1 | − 1.74·3-s + 1.05·5-s + 1.84·7-s + 0.0425·9-s + 6.02·11-s − 0.598·13-s − 1.83·15-s − 17-s + 2.57·19-s − 3.21·21-s − 5.74·23-s − 3.88·25-s + 5.15·27-s − 2.33·29-s − 4.57·31-s − 10.5·33-s + 1.94·35-s + 8.60·37-s + 1.04·39-s − 10.8·41-s − 4.58·43-s + 0.0449·45-s + 2.41·47-s − 3.60·49-s + 1.74·51-s − 2.12·53-s + 6.35·55-s + ⋯ |
| L(s) = 1 | − 1.00·3-s + 0.471·5-s + 0.696·7-s + 0.0141·9-s + 1.81·11-s − 0.166·13-s − 0.475·15-s − 0.242·17-s + 0.590·19-s − 0.701·21-s − 1.19·23-s − 0.777·25-s + 0.992·27-s − 0.433·29-s − 0.821·31-s − 1.82·33-s + 0.328·35-s + 1.41·37-s + 0.167·39-s − 1.68·41-s − 0.698·43-s + 0.00669·45-s + 0.352·47-s − 0.514·49-s + 0.244·51-s − 0.291·53-s + 0.856·55-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 | \( 1 + 1.74T + 3T^{2} \) |
| 5 | \( 1 - 1.05T + 5T^{2} \) |
| 7 | \( 1 - 1.84T + 7T^{2} \) |
| 11 | \( 1 - 6.02T + 11T^{2} \) |
| 13 | \( 1 + 0.598T + 13T^{2} \) |
| 19 | \( 1 - 2.57T + 19T^{2} \) |
| 23 | \( 1 + 5.74T + 23T^{2} \) |
| 29 | \( 1 + 2.33T + 29T^{2} \) |
| 31 | \( 1 + 4.57T + 31T^{2} \) |
| 37 | \( 1 - 8.60T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 + 4.58T + 43T^{2} \) |
| 47 | \( 1 - 2.41T + 47T^{2} \) |
| 53 | \( 1 + 2.12T + 53T^{2} \) |
| 61 | \( 1 + 0.502T + 61T^{2} \) |
| 67 | \( 1 + 8.85T + 67T^{2} \) |
| 71 | \( 1 + 7.47T + 71T^{2} \) |
| 73 | \( 1 - 2.03T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 2.18T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.30083592365129101792722945612, −6.68617136386519152818855663387, −5.90148492199974523950415182903, −5.68117987948398842397039787849, −4.66970801114370195996332147262, −4.13157301398854500204655496190, −3.17747300944606622834284709174, −1.88334977096772249294810658707, −1.33947776525169476071038213668, 0,
1.33947776525169476071038213668, 1.88334977096772249294810658707, 3.17747300944606622834284709174, 4.13157301398854500204655496190, 4.66970801114370195996332147262, 5.68117987948398842397039787849, 5.90148492199974523950415182903, 6.68617136386519152818855663387, 7.30083592365129101792722945612
