| L(s) = 1 | + 3i·2-s + 3i·3-s − 4-s − 9·6-s + 20i·7-s + 21i·8-s − 9·9-s − 24·11-s − 3i·12-s − 74i·13-s − 60·14-s − 71·16-s + 54i·17-s − 27i·18-s + 124·19-s + ⋯ |
| L(s) = 1 | + 1.06i·2-s + 0.577i·3-s − 0.125·4-s − 0.612·6-s + 1.07i·7-s + 0.928i·8-s − 0.333·9-s − 0.657·11-s − 0.0721i·12-s − 1.57i·13-s − 1.14·14-s − 1.10·16-s + 0.770i·17-s − 0.353i·18-s + 1.49·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.340841 + 1.44382i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.340841 + 1.44382i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 3iT - 8T^{2} \) |
| 7 | \( 1 - 20iT - 343T^{2} \) |
| 11 | \( 1 + 24T + 1.33e3T^{2} \) |
| 13 | \( 1 + 74iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 54iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 124T + 6.85e3T^{2} \) |
| 23 | \( 1 - 120iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 78T + 2.43e4T^{2} \) |
| 31 | \( 1 - 200T + 2.97e4T^{2} \) |
| 37 | \( 1 + 70iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 330T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 24iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 450iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 24T + 2.05e5T^{2} \) |
| 61 | \( 1 + 322T + 2.26e5T^{2} \) |
| 67 | \( 1 + 196iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 288T + 3.57e5T^{2} \) |
| 73 | \( 1 - 430iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 520T + 4.93e5T^{2} \) |
| 83 | \( 1 + 156iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.02e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 286iT - 9.12e5T^{2} \) |
| 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
−15.03125201256977245588962213195, −13.73945717343954403496584134429, −12.34464803859620993708393201461, −11.15298426659519750706286487841, −9.889490218891267285911835915667, −8.499823731422232753226216326681, −7.63597275054706864707941696511, −5.86980239748208835300091317570, −5.25187529220978039219541695512, −2.89293842875589818093107360515,
1.01638563299130841948113444224, 2.73607558073614867963940151912, 4.42027967002133456287907778075, 6.62312989728905323968477608204, 7.56623647510769697029388780433, 9.357609616257843249023703694968, 10.45087068764999500448914841418, 11.47081566805891243564620641004, 12.25758712245231039242659271315, 13.52468146302563541010408213241
