| L(s) = 1 | + 2.96·3-s + 2.90·5-s − 0.908·7-s + 5.78·9-s + 6.12·11-s − 1.06·13-s + 8.60·15-s + 17-s + 4.23·19-s − 2.69·21-s + 0.507·23-s + 3.42·25-s + 8.24·27-s − 5.65·29-s − 5.14·31-s + 18.1·33-s − 2.63·35-s − 3.29·37-s − 3.16·39-s + 7.20·41-s + 4.87·43-s + 16.7·45-s + 2.84·47-s − 6.17·49-s + 2.96·51-s − 5.19·53-s + 17.7·55-s + ⋯ |
| L(s) = 1 | + 1.71·3-s + 1.29·5-s − 0.343·7-s + 1.92·9-s + 1.84·11-s − 0.296·13-s + 2.22·15-s + 0.242·17-s + 0.970·19-s − 0.587·21-s + 0.105·23-s + 0.685·25-s + 1.58·27-s − 1.04·29-s − 0.924·31-s + 3.15·33-s − 0.445·35-s − 0.542·37-s − 0.507·39-s + 1.12·41-s + 0.743·43-s + 2.50·45-s + 0.415·47-s − 0.882·49-s + 0.414·51-s − 0.713·53-s + 2.39·55-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
| \(L(1)\) |
\(\approx\) |
\(5.956025235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.956025235\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.96T + 3T^{2} \) |
| 5 | \( 1 - 2.90T + 5T^{2} \) |
| 7 | \( 1 + 0.908T + 7T^{2} \) |
| 11 | \( 1 - 6.12T + 11T^{2} \) |
| 13 | \( 1 + 1.06T + 13T^{2} \) |
| 19 | \( 1 - 4.23T + 19T^{2} \) |
| 23 | \( 1 - 0.507T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 + 5.14T + 31T^{2} \) |
| 37 | \( 1 + 3.29T + 37T^{2} \) |
| 41 | \( 1 - 7.20T + 41T^{2} \) |
| 43 | \( 1 - 4.87T + 43T^{2} \) |
| 47 | \( 1 - 2.84T + 47T^{2} \) |
| 53 | \( 1 + 5.19T + 53T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 2.08T + 67T^{2} \) |
| 71 | \( 1 - 0.459T + 71T^{2} \) |
| 73 | \( 1 + 3.84T + 73T^{2} \) |
| 79 | \( 1 - 4.39T + 79T^{2} \) |
| 83 | \( 1 - 1.86T + 83T^{2} \) |
| 89 | \( 1 - 7.26T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−7.70321119491387839815913968188, −7.36299348332904795148956389991, −6.47518958323725291528977720367, −5.92601802259954948803651013729, −4.96773927096827524105858368339, −3.93829702399015798674495839027, −3.47016603237570549447877271482, −2.64905938260053104803990108579, −1.83288149271161650479622501754, −1.28243749154548594624472288920,
1.28243749154548594624472288920, 1.83288149271161650479622501754, 2.64905938260053104803990108579, 3.47016603237570549447877271482, 3.93829702399015798674495839027, 4.96773927096827524105858368339, 5.92601802259954948803651013729, 6.47518958323725291528977720367, 7.36299348332904795148956389991, 7.70321119491387839815913968188
