| L(s) = 1 | + 1.22·2-s − 0.534·3-s − 6.49·4-s − 20.9·5-s − 0.654·6-s + 15.0·7-s − 17.7·8-s − 26.7·9-s − 25.6·10-s + 45.4·11-s + 3.47·12-s + 3.14·13-s + 18.4·14-s + 11.2·15-s + 30.2·16-s − 32.7·18-s − 63.2·19-s + 136.·20-s − 8.03·21-s + 55.6·22-s + 114.·23-s + 9.49·24-s + 314.·25-s + 3.85·26-s + 28.7·27-s − 97.6·28-s − 96.6·29-s + ⋯ |
| L(s) = 1 | + 0.433·2-s − 0.102·3-s − 0.812·4-s − 1.87·5-s − 0.0445·6-s + 0.811·7-s − 0.784·8-s − 0.989·9-s − 0.811·10-s + 1.24·11-s + 0.0835·12-s + 0.0670·13-s + 0.351·14-s + 0.192·15-s + 0.472·16-s − 0.428·18-s − 0.763·19-s + 1.52·20-s − 0.0834·21-s + 0.539·22-s + 1.03·23-s + 0.0807·24-s + 2.51·25-s + 0.0290·26-s + 0.204·27-s − 0.659·28-s − 0.618·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.068788038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.068788038\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 1.22T + 8T^{2} \) |
| 3 | \( 1 + 0.534T + 27T^{2} \) |
| 5 | \( 1 + 20.9T + 125T^{2} \) |
| 7 | \( 1 - 15.0T + 343T^{2} \) |
| 11 | \( 1 - 45.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 3.14T + 2.19e3T^{2} \) |
| 19 | \( 1 + 63.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 114.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 96.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 194.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 73.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 341.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 281.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 36.2T + 1.03e5T^{2} \) |
| 53 | \( 1 - 191.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 104.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 517.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 560.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 378.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 877.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.19e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 783.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.60e3T + 9.12e5T^{2} \) |
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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
−11.62292686115732770586534559290, −10.83649326787360563453880119524, −9.085629736136816153992855059723, −8.511605413521685089169728401926, −7.69223027891475294621712511004, −6.34700321269914098712455093599, −4.90600437733807725950956989364, −4.19001783055156185261809165098, −3.23775744233268118687292169589, −0.68970907472670813519798433179,
0.68970907472670813519798433179, 3.23775744233268118687292169589, 4.19001783055156185261809165098, 4.90600437733807725950956989364, 6.34700321269914098712455093599, 7.69223027891475294621712511004, 8.511605413521685089169728401926, 9.085629736136816153992855059723, 10.83649326787360563453880119524, 11.62292686115732770586534559290
