| L(s) = 1 | + 1.44·2-s + 2.24·3-s + 0.0881·4-s + 3.24·6-s + 1.35·7-s − 2.76·8-s + 2.04·9-s + 4.85·11-s + 0.198·12-s + 0.198·13-s + 1.96·14-s − 4.16·16-s − 1.13·17-s + 2.96·18-s − 19-s + 3.04·21-s + 7.00·22-s + 2.55·23-s − 6.20·24-s + 0.286·26-s − 2.13·27-s + 0.119·28-s − 10.2·29-s + 2.51·31-s − 0.498·32-s + 10.8·33-s − 1.64·34-s + ⋯ |
| L(s) = 1 | + 1.02·2-s + 1.29·3-s + 0.0440·4-s + 1.32·6-s + 0.512·7-s − 0.976·8-s + 0.682·9-s + 1.46·11-s + 0.0571·12-s + 0.0549·13-s + 0.524·14-s − 1.04·16-s − 0.275·17-s + 0.697·18-s − 0.229·19-s + 0.665·21-s + 1.49·22-s + 0.532·23-s − 1.26·24-s + 0.0561·26-s − 0.411·27-s + 0.0226·28-s − 1.90·29-s + 0.451·31-s − 0.0880·32-s + 1.89·33-s − 0.281·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.204652948\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.204652948\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.44T + 2T^{2} \) |
| 3 | \( 1 - 2.24T + 3T^{2} \) |
| 7 | \( 1 - 1.35T + 7T^{2} \) |
| 11 | \( 1 - 4.85T + 11T^{2} \) |
| 13 | \( 1 - 0.198T + 13T^{2} \) |
| 17 | \( 1 + 1.13T + 17T^{2} \) |
| 23 | \( 1 - 2.55T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 2.51T + 31T^{2} \) |
| 37 | \( 1 + 0.137T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 + 7.59T + 43T^{2} \) |
| 47 | \( 1 - 2.69T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 - 5.82T + 59T^{2} \) |
| 61 | \( 1 + 7.58T + 61T^{2} \) |
| 67 | \( 1 - 8.01T + 67T^{2} \) |
| 71 | \( 1 + 8.82T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 - 3.77T + 83T^{2} \) |
| 89 | \( 1 - 9.36T + 89T^{2} \) |
| 97 | \( 1 - 0.198T + 97T^{2} \) |
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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
−11.37360375378468570780435606661, −9.885012626712489645555879755954, −8.936452791670042375929658128087, −8.605273715917640134969947995799, −7.32160862972903769704896201969, −6.26549343364618694049305406831, −5.04681577413845035166651753040, −3.97416433603683473675128906019, −3.33189380874357887448436890150, −1.93111907412621882115140510710,
1.93111907412621882115140510710, 3.33189380874357887448436890150, 3.97416433603683473675128906019, 5.04681577413845035166651753040, 6.26549343364618694049305406831, 7.32160862972903769704896201969, 8.605273715917640134969947995799, 8.936452791670042375929658128087, 9.885012626712489645555879755954, 11.37360375378468570780435606661
