| L(s) = 1 | + 1.32·2-s − 3-s − 0.231·4-s + 3.17·5-s − 1.32·6-s − 7-s − 2.96·8-s + 9-s + 4.22·10-s + 5.06·11-s + 0.231·12-s + 4.07·13-s − 1.32·14-s − 3.17·15-s − 3.48·16-s + 4.22·17-s + 1.32·18-s − 5.06·19-s − 0.735·20-s + 21-s + 6.73·22-s + 23-s + 2.96·24-s + 5.07·25-s + 5.42·26-s − 27-s + 0.231·28-s + ⋯ |
| L(s) = 1 | + 0.940·2-s − 0.577·3-s − 0.115·4-s + 1.41·5-s − 0.542·6-s − 0.377·7-s − 1.04·8-s + 0.333·9-s + 1.33·10-s + 1.52·11-s + 0.0669·12-s + 1.13·13-s − 0.355·14-s − 0.819·15-s − 0.870·16-s + 1.02·17-s + 0.313·18-s − 1.16·19-s − 0.164·20-s + 0.218·21-s + 1.43·22-s + 0.208·23-s + 0.605·24-s + 1.01·25-s + 1.06·26-s − 0.192·27-s + 0.0438·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.156147390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.156147390\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 5 | \( 1 - 3.17T + 5T^{2} \) |
| 11 | \( 1 - 5.06T + 11T^{2} \) |
| 13 | \( 1 - 4.07T + 13T^{2} \) |
| 17 | \( 1 - 4.22T + 17T^{2} \) |
| 19 | \( 1 + 5.06T + 19T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 + 2.22T + 31T^{2} \) |
| 37 | \( 1 + 1.91T + 37T^{2} \) |
| 41 | \( 1 + 1.37T + 41T^{2} \) |
| 43 | \( 1 + 3.39T + 43T^{2} \) |
| 47 | \( 1 + 5.81T + 47T^{2} \) |
| 53 | \( 1 + 6.57T + 53T^{2} \) |
| 59 | \( 1 + 5.67T + 59T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 + 2.95T + 71T^{2} \) |
| 73 | \( 1 + 2.02T + 73T^{2} \) |
| 79 | \( 1 - 7.14T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 2.08T + 97T^{2} \) |
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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
−11.09619117261514098274582636705, −10.05845502737069007787124171576, −9.336992143675788126607535629762, −8.538030128454470045880928106819, −6.51056154740179898475940086629, −6.30451106493942517353735950746, −5.41944127237424158004100042323, −4.31157475443596607822001544594, −3.24468222832909204187764363912, −1.47162342059259279481303342154,
1.47162342059259279481303342154, 3.24468222832909204187764363912, 4.31157475443596607822001544594, 5.41944127237424158004100042323, 6.30451106493942517353735950746, 6.51056154740179898475940086629, 8.538030128454470045880928106819, 9.336992143675788126607535629762, 10.05845502737069007787124171576, 11.09619117261514098274582636705
