| L(s) = 1 | − 3-s + 4·5-s − 2·9-s − 3·11-s + 5·13-s − 4·15-s + 17-s − 6·19-s − 4·23-s + 11·25-s + 5·27-s + 8·29-s + 3·33-s − 8·37-s − 5·39-s + 8·41-s + 10·43-s − 8·45-s + 2·47-s − 51-s + 3·53-s − 12·55-s + 6·57-s − 2·59-s + 8·61-s + 20·65-s + 14·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 2/3·9-s − 0.904·11-s + 1.38·13-s − 1.03·15-s + 0.242·17-s − 1.37·19-s − 0.834·23-s + 11/5·25-s + 0.962·27-s + 1.48·29-s + 0.522·33-s − 1.31·37-s − 0.800·39-s + 1.24·41-s + 1.52·43-s − 1.19·45-s + 0.291·47-s − 0.140·51-s + 0.412·53-s − 1.61·55-s + 0.794·57-s − 0.260·59-s + 1.02·61-s + 2.48·65-s + 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.004493628\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.004493628\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.592745901478006067413909081133, −8.152865808465919495805947046937, −6.71433346291073231637249834347, −6.26639307836174474805434261592, −5.65155783123102046970331334828, −5.17372327145834137805046453831, −4.02612867331051694119474885614, −2.74021814560234606035472977391, −2.10933951911823627095213212569, −0.877247665806075906579844897195,
0.877247665806075906579844897195, 2.10933951911823627095213212569, 2.74021814560234606035472977391, 4.02612867331051694119474885614, 5.17372327145834137805046453831, 5.65155783123102046970331334828, 6.26639307836174474805434261592, 6.71433346291073231637249834347, 8.152865808465919495805947046937, 8.592745901478006067413909081133
