| L(s) = 1 | − 2·2-s + 9·3-s − 28·4-s − 46·5-s − 18·6-s − 49·7-s + 120·8-s + 81·9-s + 92·10-s − 172·11-s − 252·12-s − 169·13-s + 98·14-s − 414·15-s + 656·16-s − 174·17-s − 162·18-s − 2.97e3·19-s + 1.28e3·20-s − 441·21-s + 344·22-s − 2.84e3·23-s + 1.08e3·24-s − 1.00e3·25-s + 338·26-s + 729·27-s + 1.37e3·28-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 0.577·3-s − 7/8·4-s − 0.822·5-s − 0.204·6-s − 0.377·7-s + 0.662·8-s + 1/3·9-s + 0.290·10-s − 0.428·11-s − 0.505·12-s − 0.277·13-s + 0.133·14-s − 0.475·15-s + 0.640·16-s − 0.146·17-s − 0.117·18-s − 1.88·19-s + 0.720·20-s − 0.218·21-s + 0.151·22-s − 1.12·23-s + 0.382·24-s − 0.322·25-s + 0.0980·26-s + 0.192·27-s + 0.330·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8150285137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8150285137\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 + p T + p^{5} T^{2} \) |
| 5 | \( 1 + 46 T + p^{5} T^{2} \) |
| 11 | \( 1 + 172 T + p^{5} T^{2} \) |
| 17 | \( 1 + 174 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2972 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2844 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2354 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3480 T + p^{5} T^{2} \) |
| 37 | \( 1 - 7362 T + p^{5} T^{2} \) |
| 41 | \( 1 - 8386 T + p^{5} T^{2} \) |
| 43 | \( 1 - 12476 T + p^{5} T^{2} \) |
| 47 | \( 1 - 21192 T + p^{5} T^{2} \) |
| 53 | \( 1 + 11022 T + p^{5} T^{2} \) |
| 59 | \( 1 - 38760 T + p^{5} T^{2} \) |
| 61 | \( 1 + 31070 T + p^{5} T^{2} \) |
| 67 | \( 1 - 10048 T + p^{5} T^{2} \) |
| 71 | \( 1 + 40248 T + p^{5} T^{2} \) |
| 73 | \( 1 - 4950 T + p^{5} T^{2} \) |
| 79 | \( 1 - 95072 T + p^{5} T^{2} \) |
| 83 | \( 1 - 8904 T + p^{5} T^{2} \) |
| 89 | \( 1 - 8186 T + p^{5} T^{2} \) |
| 97 | \( 1 - 158046 T + p^{5} T^{2} \) |
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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
−10.78796180116137729839600367752, −9.999252610629992256556555298655, −9.015843232387144427296351717096, −8.201403594921403799616597347224, −7.55903631912599663350727005308, −6.12098075616320405780314803299, −4.51595737993727511909072343362, −3.88807422771876662373389733862, −2.36683230689870200057217986490, −0.51852855662521543771527595900,
0.51852855662521543771527595900, 2.36683230689870200057217986490, 3.88807422771876662373389733862, 4.51595737993727511909072343362, 6.12098075616320405780314803299, 7.55903631912599663350727005308, 8.201403594921403799616597347224, 9.015843232387144427296351717096, 9.999252610629992256556555298655, 10.78796180116137729839600367752
