L(s) = 1 | − 2.26·3-s + 0.637·5-s − 2.66·7-s + 2.12·9-s − 11-s − 1.20·13-s − 1.44·15-s + 0.222·17-s + 19-s + 6.04·21-s + 9.48·23-s − 4.59·25-s + 1.97·27-s + 4.94·29-s + 3.05·31-s + 2.26·33-s − 1.70·35-s + 7.18·37-s + 2.73·39-s − 6.92·41-s − 1.53·43-s + 1.35·45-s − 5.94·47-s + 0.125·49-s − 0.502·51-s − 9.63·53-s − 0.637·55-s + ⋯ |
L(s) = 1 | − 1.30·3-s + 0.285·5-s − 1.00·7-s + 0.709·9-s − 0.301·11-s − 0.334·13-s − 0.372·15-s + 0.0538·17-s + 0.229·19-s + 1.31·21-s + 1.97·23-s − 0.918·25-s + 0.380·27-s + 0.918·29-s + 0.548·31-s + 0.394·33-s − 0.287·35-s + 1.18·37-s + 0.437·39-s − 1.08·41-s − 0.234·43-s + 0.202·45-s − 0.867·47-s + 0.0179·49-s − 0.0704·51-s − 1.32·53-s − 0.0859·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 2.26T + 3T^{2} \) |
| 5 | \( 1 - 0.637T + 5T^{2} \) |
| 7 | \( 1 + 2.66T + 7T^{2} \) |
| 13 | \( 1 + 1.20T + 13T^{2} \) |
| 17 | \( 1 - 0.222T + 17T^{2} \) |
| 23 | \( 1 - 9.48T + 23T^{2} \) |
| 29 | \( 1 - 4.94T + 29T^{2} \) |
| 31 | \( 1 - 3.05T + 31T^{2} \) |
| 37 | \( 1 - 7.18T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 1.53T + 43T^{2} \) |
| 47 | \( 1 + 5.94T + 47T^{2} \) |
| 53 | \( 1 + 9.63T + 53T^{2} \) |
| 59 | \( 1 + 2.65T + 59T^{2} \) |
| 61 | \( 1 + 0.809T + 61T^{2} \) |
| 67 | \( 1 - 2.22T + 67T^{2} \) |
| 71 | \( 1 + 2.58T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 + 5.69T + 79T^{2} \) |
| 83 | \( 1 - 2.93T + 83T^{2} \) |
| 89 | \( 1 + 5.54T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−8.205386969548162684039517674971, −7.27109688375133514993966094677, −6.45722145089256775764225397865, −6.19029754109560955963051573101, −5.13597671929187469459073753526, −4.77753439191603489589619625981, −3.43440064349344977222718886072, −2.63982007731356149131265717136, −1.13242475080857161737610997008, 0,
1.13242475080857161737610997008, 2.63982007731356149131265717136, 3.43440064349344977222718886072, 4.77753439191603489589619625981, 5.13597671929187469459073753526, 6.19029754109560955963051573101, 6.45722145089256775764225397865, 7.27109688375133514993966094677, 8.205386969548162684039517674971