| L(s) = 1 | − 0.479·2-s − 5.23·3-s − 7.77·4-s + 17.2·5-s + 2.50·6-s + 7.56·8-s + 0.376·9-s − 8.29·10-s − 32.3·11-s + 40.6·12-s + 76.6·13-s − 90.4·15-s + 58.5·16-s − 37.2·17-s − 0.180·18-s − 131.·19-s − 134.·20-s + 15.5·22-s + 77.2·23-s − 39.5·24-s + 174.·25-s − 36.7·26-s + 139.·27-s + 87.1·29-s + 43.3·30-s − 196.·31-s − 88.5·32-s + ⋯ |
| L(s) = 1 | − 0.169·2-s − 1.00·3-s − 0.971·4-s + 1.54·5-s + 0.170·6-s + 0.334·8-s + 0.0139·9-s − 0.262·10-s − 0.886·11-s + 0.978·12-s + 1.63·13-s − 1.55·15-s + 0.914·16-s − 0.532·17-s − 0.00236·18-s − 1.58·19-s − 1.50·20-s + 0.150·22-s + 0.699·23-s − 0.336·24-s + 1.39·25-s − 0.277·26-s + 0.992·27-s + 0.558·29-s + 0.264·30-s − 1.13·31-s − 0.489·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.139855873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.139855873\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 47 | \( 1 - 47T \) |
| good | 2 | \( 1 + 0.479T + 8T^{2} \) |
| 3 | \( 1 + 5.23T + 27T^{2} \) |
| 5 | \( 1 - 17.2T + 125T^{2} \) |
| 11 | \( 1 + 32.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 37.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 131.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 77.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 87.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 196.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 270.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 387.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 398.T + 7.95e4T^{2} \) |
| 53 | \( 1 + 452.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 148.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 846.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 490.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 316.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.12e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 916.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 746.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 470.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 206.T + 9.12e5T^{2} \) |
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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
−8.717187234692623800748119809400, −8.208284355073422287139045915385, −6.75736189568509290100776119007, −6.12321232927086062679364887971, −5.56554960965767267587176475935, −4.95087094949682364655088035701, −4.00040022502607495558890465392, −2.67490920833490431275993335052, −1.54909467810848855383214148702, −0.54278362424666438376798887049,
0.54278362424666438376798887049, 1.54909467810848855383214148702, 2.67490920833490431275993335052, 4.00040022502607495558890465392, 4.95087094949682364655088035701, 5.56554960965767267587176475935, 6.12321232927086062679364887971, 6.75736189568509290100776119007, 8.208284355073422287139045915385, 8.717187234692623800748119809400
