L(s) = 1 | + (−1 − i)2-s + (1.22 − 1.22i)3-s + 2i·4-s + (3.24 + 3.80i)5-s − 2.44·6-s + (−6.75 − 6.75i)7-s + (2 − 2i)8-s − 2.99i·9-s + (0.558 − 7.04i)10-s + 21.5·11-s + (2.44 + 2.44i)12-s + (−1.08 + 1.08i)13-s + 13.5i·14-s + (8.63 + 0.683i)15-s − 4·16-s + (4.33 + 4.33i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.5i)2-s + (0.408 − 0.408i)3-s + 0.5i·4-s + (0.649 + 0.760i)5-s − 0.408·6-s + (−0.965 − 0.965i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.0558 − 0.704i)10-s + 1.96·11-s + (0.204 + 0.204i)12-s + (−0.0833 + 0.0833i)13-s + 0.965i·14-s + (0.575 + 0.0455i)15-s − 0.250·16-s + (0.255 + 0.255i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.456 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.824801803\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.824801803\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
| 5 | \( 1 + (-3.24 - 3.80i)T \) |
| 23 | \( 1 + (3.39 - 3.39i)T \) |
good | 7 | \( 1 + (6.75 + 6.75i)T + 49iT^{2} \) |
| 11 | \( 1 - 21.5T + 121T^{2} \) |
| 13 | \( 1 + (1.08 - 1.08i)T - 169iT^{2} \) |
| 17 | \( 1 + (-4.33 - 4.33i)T + 289iT^{2} \) |
| 19 | \( 1 - 1.61iT - 361T^{2} \) |
| 29 | \( 1 + 10.6iT - 841T^{2} \) |
| 31 | \( 1 - 21.7T + 961T^{2} \) |
| 37 | \( 1 + (18.5 + 18.5i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 55.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-45.1 + 45.1i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (40.0 + 40.0i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (22.8 - 22.8i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + 45.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 106.T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-62.2 - 62.2i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 95.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (18.3 - 18.3i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 74.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-29.2 + 29.2i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 148. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (47.9 + 47.9i)T + 9.40e3iT^{2} \) |
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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
−9.869524129741454385269254047901, −9.548438927190368686591599886140, −8.587109747034695233847275781197, −7.30915440815686520635141151041, −6.78278966562956257992567557905, −6.04010661873619556912036713926, −3.99983257253082902491681263060, −3.42009132969405996650354103715, −2.12204371471815755908393958734, −0.908515463464929145067956605758,
1.15739163455115091120975908912, 2.57318965090937561454850606848, 3.95867563091618329492405713762, 5.11777407143196241202985416653, 6.13919971457831350385874874726, 6.66228388216221250864427756953, 8.121631337569955766668668045247, 8.916678800557398350843599545351, 9.523011358106763131545788981878, 9.744886307789779697778157101555