| L(s) = 1 | + 1.26·2-s − 6.39·4-s − 3.92·5-s + 7·7-s − 18.2·8-s − 4.98·10-s + 51.7·11-s + 67.5·13-s + 8.87·14-s + 27.9·16-s + 63.4·17-s − 71.9·19-s + 25.1·20-s + 65.5·22-s + 147.·23-s − 109.·25-s + 85.6·26-s − 44.7·28-s + 117.·29-s + 54.7·31-s + 181.·32-s + 80.5·34-s − 27.4·35-s + 9.70·37-s − 91.1·38-s + 71.6·40-s − 236.·41-s + ⋯ |
| L(s) = 1 | + 0.448·2-s − 0.799·4-s − 0.351·5-s + 0.377·7-s − 0.806·8-s − 0.157·10-s + 1.41·11-s + 1.44·13-s + 0.169·14-s + 0.437·16-s + 0.905·17-s − 0.868·19-s + 0.280·20-s + 0.635·22-s + 1.33·23-s − 0.876·25-s + 0.646·26-s − 0.302·28-s + 0.753·29-s + 0.316·31-s + 1.00·32-s + 0.406·34-s − 0.132·35-s + 0.0431·37-s − 0.389·38-s + 0.283·40-s − 0.900·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.890009273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.890009273\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 - 1.26T + 8T^{2} \) |
| 5 | \( 1 + 3.92T + 125T^{2} \) |
| 11 | \( 1 - 51.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 67.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 63.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 71.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 147.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 117.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 54.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 9.70T + 5.06e4T^{2} \) |
| 41 | \( 1 + 236.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 489.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 613.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 316.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 2.43T + 2.05e5T^{2} \) |
| 61 | \( 1 - 482.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 646.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 459.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 137.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 816.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 255.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 62.9T + 9.12e5T^{2} \) |
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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
−12.11050418120095162927090243564, −11.38877843987926079005848866597, −10.11273323659505284049996330443, −8.883230783940692852695745409819, −8.342258761951479821104636953239, −6.73961172359149170084649142515, −5.62749816620686311188033767704, −4.30097411355325161304113297813, −3.48890578831053268065585151651, −1.10424273914690404018298632997,
1.10424273914690404018298632997, 3.48890578831053268065585151651, 4.30097411355325161304113297813, 5.62749816620686311188033767704, 6.73961172359149170084649142515, 8.342258761951479821104636953239, 8.883230783940692852695745409819, 10.11273323659505284049996330443, 11.38877843987926079005848866597, 12.11050418120095162927090243564
