L(s) = 1 | + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (−1.12 − 1.40i)5-s + (−0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (0.400 − 1.75i)10-s + (0.400 + 0.193i)13-s + (−0.900 − 0.433i)16-s + 1.24·17-s + (−0.900 − 0.433i)18-s + (1.62 − 0.781i)20-s + (−0.500 + 2.19i)25-s + (0.0990 + 0.433i)26-s + (−0.900 − 0.433i)29-s + (−0.222 − 0.974i)32-s + ⋯ |
L(s) = 1 | + (0.623 + 0.781i)2-s + (−0.222 + 0.974i)4-s + (−1.12 − 1.40i)5-s + (−0.900 + 0.433i)8-s + (−0.900 + 0.433i)9-s + (0.400 − 1.75i)10-s + (0.400 + 0.193i)13-s + (−0.900 − 0.433i)16-s + 1.24·17-s + (−0.900 − 0.433i)18-s + (1.62 − 0.781i)20-s + (−0.500 + 2.19i)25-s + (0.0990 + 0.433i)26-s + (−0.900 − 0.433i)29-s + (−0.222 − 0.974i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.724 - 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6567988611\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6567988611\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 29 | \( 1 + (0.900 + 0.433i)T \) |
good | 3 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 5 | \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \) |
| 7 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 11 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 13 | \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 - 1.24T + T^{2} \) |
| 19 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 23 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 37 | \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \) |
| 41 | \( 1 + 0.445T + T^{2} \) |
| 43 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 47 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 53 | \( 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \) |
| 67 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \) |
| 79 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 83 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \) |
| 97 | \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \) |
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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
−13.93728280698268691617559719340, −12.92393525514533243599791940597, −12.08547159155992900167548059065, −11.31463289792714068287782596293, −9.209130155494114159252985183962, −8.262113482893172886022719944456, −7.61735796559339943625723157582, −5.80485425628311094218152912818, −4.81320037843895081907449787425, −3.57233705586428297510806529680,
2.97731371320272861811544350296, 3.78211432159173617962086038358, 5.66283865437418854723733575503, 6.89243826915145120674413000352, 8.279589527838960030130262015903, 9.852576512744813937403095779817, 10.93232029820198115034421540498, 11.53572814064135719676854503464, 12.37507573823034148052968421453, 13.79756665727488847040209539682