| L(s) = 1 | + 2-s − 1.87·3-s + 4-s + 3.85·5-s − 1.87·6-s + 7-s + 8-s + 0.528·9-s + 3.85·10-s − 1.86·11-s − 1.87·12-s + 5.20·13-s + 14-s − 7.24·15-s + 16-s + 6.37·17-s + 0.528·18-s − 7.67·19-s + 3.85·20-s − 1.87·21-s − 1.86·22-s + 2.91·23-s − 1.87·24-s + 9.86·25-s + 5.20·26-s + 4.64·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.08·3-s + 0.5·4-s + 1.72·5-s − 0.766·6-s + 0.377·7-s + 0.353·8-s + 0.176·9-s + 1.21·10-s − 0.562·11-s − 0.542·12-s + 1.44·13-s + 0.267·14-s − 1.87·15-s + 0.250·16-s + 1.54·17-s + 0.124·18-s − 1.76·19-s + 0.862·20-s − 0.409·21-s − 0.397·22-s + 0.607·23-s − 0.383·24-s + 1.97·25-s + 1.02·26-s + 0.893·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1106 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1106 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.509248646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.509248646\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 3 | \( 1 + 1.87T + 3T^{2} \) |
| 5 | \( 1 - 3.85T + 5T^{2} \) |
| 11 | \( 1 + 1.86T + 11T^{2} \) |
| 13 | \( 1 - 5.20T + 13T^{2} \) |
| 17 | \( 1 - 6.37T + 17T^{2} \) |
| 19 | \( 1 + 7.67T + 19T^{2} \) |
| 23 | \( 1 - 2.91T + 23T^{2} \) |
| 29 | \( 1 + 4.62T + 29T^{2} \) |
| 31 | \( 1 + 8.49T + 31T^{2} \) |
| 37 | \( 1 + 0.886T + 37T^{2} \) |
| 41 | \( 1 + 1.01T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + 8.86T + 47T^{2} \) |
| 53 | \( 1 - 3.13T + 53T^{2} \) |
| 59 | \( 1 - 11.0T + 59T^{2} \) |
| 61 | \( 1 + 2.67T + 61T^{2} \) |
| 67 | \( 1 - 9.04T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 7.10T + 73T^{2} \) |
| 83 | \( 1 - 1.00T + 83T^{2} \) |
| 89 | \( 1 - 2.73T + 89T^{2} \) |
| 97 | \( 1 + 0.922T + 97T^{2} \) |
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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
−10.16931614368676527023099062235, −9.144478712244280570196112734451, −8.197717072949359439138857565996, −6.88380382280969139234265270810, −6.09217174973988951471063872535, −5.60099510300601528732602889249, −5.12461122588789251667774968273, −3.75167633633478920127177629537, −2.38151127712076401153359836851, −1.29925587636223275804576431778,
1.29925587636223275804576431778, 2.38151127712076401153359836851, 3.75167633633478920127177629537, 5.12461122588789251667774968273, 5.60099510300601528732602889249, 6.09217174973988951471063872535, 6.88380382280969139234265270810, 8.197717072949359439138857565996, 9.144478712244280570196112734451, 10.16931614368676527023099062235
