| L(s) = 1 | + 2.24·2-s + 3-s + 3.03·4-s + 0.599·5-s + 2.24·6-s + 4.24·7-s + 2.32·8-s + 9-s + 1.34·10-s + 3.03·12-s − 13-s + 9.51·14-s + 0.599·15-s − 0.852·16-s + 3.82·17-s + 2.24·18-s + 5.08·19-s + 1.81·20-s + 4.24·21-s − 8.18·23-s + 2.32·24-s − 4.64·25-s − 2.24·26-s + 27-s + 12.8·28-s + 5.00·29-s + 1.34·30-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 0.577·3-s + 1.51·4-s + 0.268·5-s + 0.916·6-s + 1.60·7-s + 0.822·8-s + 0.333·9-s + 0.425·10-s + 0.876·12-s − 0.277·13-s + 2.54·14-s + 0.154·15-s − 0.213·16-s + 0.927·17-s + 0.528·18-s + 1.16·19-s + 0.406·20-s + 0.925·21-s − 1.70·23-s + 0.474·24-s − 0.928·25-s − 0.440·26-s + 0.192·27-s + 2.43·28-s + 0.928·29-s + 0.245·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4719 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.822940322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.822940322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.24T + 2T^{2} \) |
| 5 | \( 1 - 0.599T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 17 | \( 1 - 3.82T + 17T^{2} \) |
| 19 | \( 1 - 5.08T + 19T^{2} \) |
| 23 | \( 1 + 8.18T + 23T^{2} \) |
| 29 | \( 1 - 5.00T + 29T^{2} \) |
| 31 | \( 1 - 4.79T + 31T^{2} \) |
| 37 | \( 1 + 1.22T + 37T^{2} \) |
| 41 | \( 1 + 7.87T + 41T^{2} \) |
| 43 | \( 1 - 7.54T + 43T^{2} \) |
| 47 | \( 1 + 4.90T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 2.91T + 59T^{2} \) |
| 61 | \( 1 + 5.50T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 8.14T + 73T^{2} \) |
| 79 | \( 1 + 9.23T + 79T^{2} \) |
| 83 | \( 1 + 0.172T + 83T^{2} \) |
| 89 | \( 1 + 7.65T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.025233049066131389969260105427, −7.61559334426814723354262265934, −6.71703179902989581774037746868, −5.69797736927207845240091768984, −5.35312106701078905172953772750, −4.49053576519580921524540258258, −3.96715499941070748609610864394, −3.01577071914860625970007785543, −2.20906473150044869137284492248, −1.38783604197327974492164826229,
1.38783604197327974492164826229, 2.20906473150044869137284492248, 3.01577071914860625970007785543, 3.96715499941070748609610864394, 4.49053576519580921524540258258, 5.35312106701078905172953772750, 5.69797736927207845240091768984, 6.71703179902989581774037746868, 7.61559334426814723354262265934, 8.025233049066131389969260105427
