| L(s) = 1 | + 0.364·2-s + 3-s − 1.86·4-s − 0.530·5-s + 0.364·6-s + 4.57·7-s − 1.40·8-s + 9-s − 0.193·10-s − 1.08·11-s − 1.86·12-s + 4.51·13-s + 1.66·14-s − 0.530·15-s + 3.22·16-s − 1.79·17-s + 0.364·18-s − 1.25·19-s + 0.990·20-s + 4.57·21-s − 0.395·22-s + 6.69·23-s − 1.40·24-s − 4.71·25-s + 1.64·26-s + 27-s − 8.54·28-s + ⋯ |
| L(s) = 1 | + 0.257·2-s + 0.577·3-s − 0.933·4-s − 0.237·5-s + 0.148·6-s + 1.72·7-s − 0.498·8-s + 0.333·9-s − 0.0611·10-s − 0.327·11-s − 0.539·12-s + 1.25·13-s + 0.445·14-s − 0.136·15-s + 0.805·16-s − 0.435·17-s + 0.0858·18-s − 0.288·19-s + 0.221·20-s + 0.998·21-s − 0.0843·22-s + 1.39·23-s − 0.287·24-s − 0.943·25-s + 0.322·26-s + 0.192·27-s − 1.61·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 453 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.801610054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.801610054\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 151 | \( 1 + T \) |
| good | 2 | \( 1 - 0.364T + 2T^{2} \) |
| 5 | \( 1 + 0.530T + 5T^{2} \) |
| 7 | \( 1 - 4.57T + 7T^{2} \) |
| 11 | \( 1 + 1.08T + 11T^{2} \) |
| 13 | \( 1 - 4.51T + 13T^{2} \) |
| 17 | \( 1 + 1.79T + 17T^{2} \) |
| 19 | \( 1 + 1.25T + 19T^{2} \) |
| 23 | \( 1 - 6.69T + 23T^{2} \) |
| 29 | \( 1 - 8.35T + 29T^{2} \) |
| 31 | \( 1 - 2.38T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 8.49T + 43T^{2} \) |
| 47 | \( 1 + 6.83T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 2.49T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 + 1.43T + 67T^{2} \) |
| 71 | \( 1 + 9.01T + 71T^{2} \) |
| 73 | \( 1 + 3.82T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 + 8.02T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 1.55T + 97T^{2} \) |
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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
−11.15635781733209412280250469012, −10.19585226993160138566089468851, −8.988399621763335275747146946892, −8.328369899882356010453311073054, −7.86617284189815229686115419852, −6.32100608279491301747541660392, −4.94118548019958435928551353502, −4.44514386050797559619362575452, −3.17134846586109404510741403775, −1.42245055119170924841242643473,
1.42245055119170924841242643473, 3.17134846586109404510741403775, 4.44514386050797559619362575452, 4.94118548019958435928551353502, 6.32100608279491301747541660392, 7.86617284189815229686115419852, 8.328369899882356010453311073054, 8.988399621763335275747146946892, 10.19585226993160138566089468851, 11.15635781733209412280250469012
