| L(s) = 1 | + 5.19i·3-s + 8·4-s + (−10 − 15.5i)7-s − 27·9-s + 41.5i·12-s − 62.3i·13-s + 64·16-s + 155. i·19-s + (81 − 51.9i)21-s − 125·25-s − 140. i·27-s + (−80 − 124. i)28-s + 155. i·31-s − 216·36-s − 110·37-s + ⋯ |
| L(s) = 1 | + 0.999i·3-s + 4-s + (−0.539 − 0.841i)7-s − 9-s + 0.999i·12-s − 1.33i·13-s + 16-s + 1.88i·19-s + (0.841 − 0.539i)21-s − 25-s − 1.00i·27-s + (−0.539 − 0.841i)28-s + 0.903i·31-s − 36-s − 0.488·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.15884 + 0.339751i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.15884 + 0.339751i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.19iT \) |
| 7 | \( 1 + (10 + 15.5i)T \) |
| good | 2 | \( 1 - 8T^{2} \) |
| 5 | \( 1 + 125T^{2} \) |
| 11 | \( 1 - 1.33e3T^{2} \) |
| 13 | \( 1 + 62.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 4.91e3T^{2} \) |
| 19 | \( 1 - 155. iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 1.21e4T^{2} \) |
| 29 | \( 1 - 2.43e4T^{2} \) |
| 31 | \( 1 - 155. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 110T + 5.06e4T^{2} \) |
| 41 | \( 1 + 6.89e4T^{2} \) |
| 43 | \( 1 - 520T + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 - 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 + 935. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 880T + 3.00e5T^{2} \) |
| 71 | \( 1 - 3.57e5T^{2} \) |
| 73 | \( 1 - 374. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 884T + 4.93e5T^{2} \) |
| 83 | \( 1 + 5.71e5T^{2} \) |
| 89 | \( 1 + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.37e3iT - 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.38982325996438439646512944928, −16.34500091158247461705829394231, −15.52454943668811818083084724822, −14.23362462005558988849310461140, −12.37696182031449400309338652648, −10.77166038856977473910419787828, −10.00703873359342517455136202866, −7.83476027603686869972545400194, −5.89454847868845187990304039147, −3.48417398160779976635838678149,
2.39575046885036427936358363073, 6.09572951414993823770614548241, 7.24623828832099096945840415539, 9.084401795919869858826569941069, 11.30499475818402045982129225752, 12.16007576358184998281479011544, 13.53575954062218435451689079563, 15.11259072625038686657331790189, 16.30485985708187478834088296120, 17.63527759404419863783128520334
