L(s) = 1 | − 5.09·2-s + (5.00 − 1.41i)3-s + 17.9·4-s + (−3.42 − 5.92i)5-s + (−25.4 + 7.19i)6-s + (−0.241 + 18.5i)7-s − 50.6·8-s + (23.0 − 14.1i)9-s + (17.4 + 30.1i)10-s + (20.5 − 35.5i)11-s + (89.7 − 25.3i)12-s + (31.8 − 55.2i)13-s + (1.23 − 94.3i)14-s + (−25.4 − 24.7i)15-s + 114.·16-s + (38.0 + 65.8i)17-s + ⋯ |
L(s) = 1 | − 1.80·2-s + (0.962 − 0.271i)3-s + 2.24·4-s + (−0.305 − 0.529i)5-s + (−1.73 + 0.489i)6-s + (−0.0130 + 0.999i)7-s − 2.24·8-s + (0.852 − 0.523i)9-s + (0.550 + 0.954i)10-s + (0.562 − 0.974i)11-s + (2.15 − 0.610i)12-s + (0.680 − 1.17i)13-s + (0.0234 − 1.80i)14-s + (−0.438 − 0.426i)15-s + 1.79·16-s + (0.542 + 0.939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.700 + 0.714i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.700 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.807169 - 0.339012i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.807169 - 0.339012i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-5.00 + 1.41i)T \) |
| 7 | \( 1 + (0.241 - 18.5i)T \) |
good | 2 | \( 1 + 5.09T + 8T^{2} \) |
| 5 | \( 1 + (3.42 + 5.92i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-20.5 + 35.5i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-31.8 + 55.2i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-38.0 - 65.8i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-46.7 + 81.0i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (21.0 + 36.4i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-21.9 - 37.9i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 200.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (52.1 - 90.3i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (108. - 188. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-236. - 409. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + 17.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-211. - 365. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 - 291.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 154.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 838.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 940.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-65.8 - 113. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 - 620.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-102. - 177. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-432. + 749. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-331. - 573. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
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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
−14.74955233962544634312042617316, −13.00780017057069619084795594791, −11.84844165708937765702821450656, −10.54877562783033631043680035414, −9.164174156331377132840473641902, −8.586053419531365800079653903585, −7.81944505481407992123651475750, −6.17797512043969974230808217003, −3.00872075705702476697106215035, −1.12197241959487120018474436019,
1.67470356833191887468302229357, 3.71893228104131027220757583891, 7.08141732409914285412513103135, 7.47297311290560603970999561905, 8.947314006948785413252085394733, 9.770186687469928910759546698522, 10.67773457024261715842107377074, 11.86752190372582382647966389887, 13.85240144363301874954472356179, 14.80893809827326768907817870576