| L(s) = 1 | − 2.47i·2-s + (−2.89 + 0.771i)3-s − 2.11·4-s + 6.48i·5-s + (1.90 + 7.16i)6-s − 2.64·7-s − 4.66i·8-s + (7.80 − 4.47i)9-s + 16.0·10-s − 9.37i·11-s + (6.13 − 1.63i)12-s + 4.90·13-s + 6.54i·14-s + (−5.00 − 18.8i)15-s − 19.9·16-s + 11.1i·17-s + ⋯ |
| L(s) = 1 | − 1.23i·2-s + (−0.966 + 0.257i)3-s − 0.528·4-s + 1.29i·5-s + (0.318 + 1.19i)6-s − 0.377·7-s − 0.582i·8-s + (0.867 − 0.497i)9-s + 1.60·10-s − 0.851i·11-s + (0.510 − 0.136i)12-s + 0.377·13-s + 0.467i·14-s + (−0.333 − 1.25i)15-s − 1.24·16-s + 0.653i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2026884796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2026884796\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.89 - 0.771i)T \) |
| 7 | \( 1 + 2.64T \) |
| 23 | \( 1 + 4.79iT \) |
| good | 2 | \( 1 + 2.47iT - 4T^{2} \) |
| 5 | \( 1 - 6.48iT - 25T^{2} \) |
| 11 | \( 1 + 9.37iT - 121T^{2} \) |
| 13 | \( 1 - 4.90T + 169T^{2} \) |
| 17 | \( 1 - 11.1iT - 289T^{2} \) |
| 19 | \( 1 + 6.35T + 361T^{2} \) |
| 29 | \( 1 - 49.3iT - 841T^{2} \) |
| 31 | \( 1 + 46.1T + 961T^{2} \) |
| 37 | \( 1 + 65.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 44.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 77.9T + 1.84e3T^{2} \) |
| 47 | \( 1 + 2.11iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 66.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 69.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 22.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 85.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 45.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 108.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 42.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 143. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 35.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 77.3T + 9.40e3T^{2} \) |
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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
−11.02014283536273599894966633418, −10.49219435886218824839162098148, −9.840311808099481281522125978906, −8.634559733294552810799639273686, −6.90265870722073234982572785421, −6.57176705675742764359945493577, −5.33640811974672798673265806023, −3.72758741473529871615913429996, −3.22888767601144028121540634518, −1.64044326105399131875710877842,
0.090542807069770423094835278049, 1.81638907968274679241059219797, 4.26401991057290853411026596281, 5.15591886058020298088278849152, 5.77857397564759936570379063891, 6.83098984718446276901410386645, 7.49600993337087263571714460469, 8.538276789556085950465975074390, 9.386042480035414574335116644685, 10.46367241587395810086740898912
