| L(s) = 1 | + 2.92·2-s + 10.0·3-s + 0.531·4-s + 29.3·6-s − 21.8·8-s + 73.9·9-s + 15.5·11-s + 5.34·12-s + 63.9·13-s − 67.9·16-s − 10.5·17-s + 216.·18-s − 38.7·19-s + 45.5·22-s + 24.1·23-s − 219.·24-s + 186.·26-s + 471.·27-s + 226.·29-s − 143.·31-s − 24.0·32-s + 156.·33-s − 30.8·34-s + 39.3·36-s + 285.·37-s − 113.·38-s + 642.·39-s + ⋯ |
| L(s) = 1 | + 1.03·2-s + 1.93·3-s + 0.0664·4-s + 1.99·6-s − 0.964·8-s + 2.73·9-s + 0.427·11-s + 0.128·12-s + 1.36·13-s − 1.06·16-s − 0.150·17-s + 2.82·18-s − 0.468·19-s + 0.441·22-s + 0.218·23-s − 1.86·24-s + 1.40·26-s + 3.36·27-s + 1.45·29-s − 0.829·31-s − 0.132·32-s + 0.826·33-s − 0.155·34-s + 0.182·36-s + 1.26·37-s − 0.483·38-s + 2.63·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.650008317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.650008317\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.92T + 8T^{2} \) |
| 3 | \( 1 - 10.0T + 27T^{2} \) |
| 11 | \( 1 - 15.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 63.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 10.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 38.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 24.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 226.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 143.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 285.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 32.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 235.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 144.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 318.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 448.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 166.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 741.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 373.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 237.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 465.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 691.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−9.047939591325998864698279600724, −8.682044067126025579618915670401, −7.898005499050033655348637317073, −6.81835910989726860568250281310, −6.02379423443828374270026522483, −4.60683542846319880682336499815, −4.01352575741614710626749266346, −3.28684051836335433364881879206, −2.49123241130024186202853031245, −1.23841016273211106927780576375,
1.23841016273211106927780576375, 2.49123241130024186202853031245, 3.28684051836335433364881879206, 4.01352575741614710626749266346, 4.60683542846319880682336499815, 6.02379423443828374270026522483, 6.81835910989726860568250281310, 7.898005499050033655348637317073, 8.682044067126025579618915670401, 9.047939591325998864698279600724
