| L(s) = 1 | + 0.937·2-s − 1.12·4-s + 3.32·5-s − 3.23·7-s − 2.92·8-s + 3.11·10-s − 0.334·11-s + 2.81·13-s − 3.02·14-s − 0.504·16-s + 0.999·17-s + 2.23·19-s − 3.72·20-s − 0.313·22-s − 5.04·23-s + 6.04·25-s + 2.63·26-s + 3.61·28-s − 5.55·29-s + 5.38·32-s + 0.937·34-s − 10.7·35-s − 0.688·37-s + 2.09·38-s − 9.72·40-s − 9.91·41-s + 12.5·43-s + ⋯ |
| L(s) = 1 | + 0.663·2-s − 0.560·4-s + 1.48·5-s − 1.22·7-s − 1.03·8-s + 0.985·10-s − 0.100·11-s + 0.780·13-s − 0.809·14-s − 0.126·16-s + 0.242·17-s + 0.511·19-s − 0.832·20-s − 0.0669·22-s − 1.05·23-s + 1.20·25-s + 0.517·26-s + 0.683·28-s − 1.03·29-s + 0.951·32-s + 0.160·34-s − 1.81·35-s − 0.113·37-s + 0.339·38-s − 1.53·40-s − 1.54·41-s + 1.91·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.499785407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.499785407\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 0.937T + 2T^{2} \) |
| 5 | \( 1 - 3.32T + 5T^{2} \) |
| 7 | \( 1 + 3.23T + 7T^{2} \) |
| 11 | \( 1 + 0.334T + 11T^{2} \) |
| 13 | \( 1 - 2.81T + 13T^{2} \) |
| 17 | \( 1 - 0.999T + 17T^{2} \) |
| 19 | \( 1 - 2.23T + 19T^{2} \) |
| 23 | \( 1 + 5.04T + 23T^{2} \) |
| 29 | \( 1 + 5.55T + 29T^{2} \) |
| 37 | \( 1 + 0.688T + 37T^{2} \) |
| 41 | \( 1 + 9.91T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 9.95T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 - 3.14T + 59T^{2} \) |
| 61 | \( 1 + 10.8T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 - 4.80T + 71T^{2} \) |
| 73 | \( 1 + 7.96T + 73T^{2} \) |
| 79 | \( 1 - 2.26T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.69566795056048282310556674639, −6.78572686556242205258151131084, −6.04451929063226869095774122957, −5.79841164809236815605076131976, −5.22269348985386035592799306752, −4.13922819802384305433462784282, −3.54870052354031076456147235479, −2.79612207585869750338411686306, −1.92301068286512047855840174243, −0.68916490817302971549409737206,
0.68916490817302971549409737206, 1.92301068286512047855840174243, 2.79612207585869750338411686306, 3.54870052354031076456147235479, 4.13922819802384305433462784282, 5.22269348985386035592799306752, 5.79841164809236815605076131976, 6.04451929063226869095774122957, 6.78572686556242205258151131084, 7.69566795056048282310556674639
