| L(s) = 1 | − 1.21i·2-s + (−2.46 + 8.65i)3-s + 14.5·4-s − 2.20i·5-s + (10.5 + 2.99i)6-s + 82.0·7-s − 37.0i·8-s + (−68.8 − 42.6i)9-s − 2.67·10-s − 214. i·11-s + (−35.8 + 125. i)12-s + 64.5·13-s − 99.6i·14-s + (19.0 + 5.43i)15-s + 187.·16-s + 71.0i·17-s + ⋯ |
| L(s) = 1 | − 0.303i·2-s + (−0.274 + 0.961i)3-s + 0.907·4-s − 0.0881i·5-s + (0.292 + 0.0832i)6-s + 1.67·7-s − 0.579i·8-s + (−0.849 − 0.527i)9-s − 0.0267·10-s − 1.77i·11-s + (−0.248 + 0.872i)12-s + 0.382·13-s − 0.508i·14-s + (0.0847 + 0.0241i)15-s + 0.731·16-s + 0.245i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.498778165\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.498778165\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.46 - 8.65i)T \) |
| 59 | \( 1 + 453. iT \) |
| good | 2 | \( 1 + 1.21iT - 16T^{2} \) |
| 5 | \( 1 + 2.20iT - 625T^{2} \) |
| 7 | \( 1 - 82.0T + 2.40e3T^{2} \) |
| 11 | \( 1 + 214. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 64.5T + 2.85e4T^{2} \) |
| 17 | \( 1 - 71.0iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 346.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 647. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 909. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 115.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 22.9T + 1.87e6T^{2} \) |
| 41 | \( 1 - 830. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.66e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.42e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.37e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 + 691.T + 1.38e7T^{2} \) |
| 67 | \( 1 - 6.32e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.99e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 9.27e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.17e4T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.00e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 7.60e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 4.99e3T + 8.85e7T^{2} \) |
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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
−11.49250603298848276521984716655, −10.94921643748499942386139311351, −10.55251098773705964595207984289, −8.772066711531036906484476367539, −8.199876581717145349983897511356, −6.47179901954166976988678168264, −5.43956932175983572101467311876, −4.18761021899607897981025797748, −2.81392299697429403275577332276, −1.07453205037585881452447103004,
1.51593073125332031158162905382, 2.27804761116120020271155697476, 4.66473116973334638278466585975, 5.80385277588932311695680135844, 7.09582158285827723712860714245, 7.58601312975481486283530081521, 8.590265142673669424842062220280, 10.38666660197395178460154929447, 11.33562375506571308071231093084, 11.89992444689500054161322845691
