| L(s) = 1 | + 0.804·3-s + 3.90·5-s + 3.10·7-s − 2.35·9-s + 2.34·11-s + 5.34·13-s + 3.14·15-s + 17-s + 2.48·19-s + 2.49·21-s − 4.84·23-s + 10.2·25-s − 4.30·27-s − 6.16·29-s − 1.04·31-s + 1.88·33-s + 12.1·35-s − 2.50·37-s + 4.30·39-s + 0.210·41-s + 6.66·43-s − 9.19·45-s + 1.21·47-s + 2.64·49-s + 0.804·51-s + 6.60·53-s + 9.17·55-s + ⋯ |
| L(s) = 1 | + 0.464·3-s + 1.74·5-s + 1.17·7-s − 0.784·9-s + 0.708·11-s + 1.48·13-s + 0.811·15-s + 0.242·17-s + 0.570·19-s + 0.545·21-s − 1.00·23-s + 2.05·25-s − 0.828·27-s − 1.14·29-s − 0.187·31-s + 0.328·33-s + 2.05·35-s − 0.411·37-s + 0.688·39-s + 0.0328·41-s + 1.01·43-s − 1.37·45-s + 0.177·47-s + 0.377·49-s + 0.112·51-s + 0.907·53-s + 1.23·55-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
| \(L(1)\) |
\(\approx\) |
\(4.548214587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.548214587\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.804T + 3T^{2} \) |
| 5 | \( 1 - 3.90T + 5T^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 - 2.34T + 11T^{2} \) |
| 13 | \( 1 - 5.34T + 13T^{2} \) |
| 19 | \( 1 - 2.48T + 19T^{2} \) |
| 23 | \( 1 + 4.84T + 23T^{2} \) |
| 29 | \( 1 + 6.16T + 29T^{2} \) |
| 31 | \( 1 + 1.04T + 31T^{2} \) |
| 37 | \( 1 + 2.50T + 37T^{2} \) |
| 41 | \( 1 - 0.210T + 41T^{2} \) |
| 43 | \( 1 - 6.66T + 43T^{2} \) |
| 47 | \( 1 - 1.21T + 47T^{2} \) |
| 53 | \( 1 - 6.60T + 53T^{2} \) |
| 61 | \( 1 - 4.17T + 61T^{2} \) |
| 67 | \( 1 - 3.77T + 67T^{2} \) |
| 71 | \( 1 + 8.74T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 2.63T + 79T^{2} \) |
| 83 | \( 1 + 3.25T + 83T^{2} \) |
| 89 | \( 1 + 2.96T + 89T^{2} \) |
| 97 | \( 1 + 1.64T + 97T^{2} \) |
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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
−8.087553755044770017566225134831, −7.10642580004828999415311425055, −6.26748445721004280000899648994, −5.60434623006726044164691236000, −5.43896747619976948036905642432, −4.20803355330794997620300738282, −3.47933908332646010118032637867, −2.45241194948676783463274871527, −1.78014381372204614361821796669, −1.16149502763182650320516020977,
1.16149502763182650320516020977, 1.78014381372204614361821796669, 2.45241194948676783463274871527, 3.47933908332646010118032637867, 4.20803355330794997620300738282, 5.43896747619976948036905642432, 5.60434623006726044164691236000, 6.26748445721004280000899648994, 7.10642580004828999415311425055, 8.087553755044770017566225134831
