L(s) = 1 | + i·2-s − 4-s − 2.99i·5-s + (−0.713 + 2.54i)7-s − i·8-s + 2.99·10-s − 1.70·11-s + 1.66·13-s + (−2.54 − 0.713i)14-s + 16-s − 4.46i·17-s + (2.13 + 3.79i)19-s + 2.99i·20-s − 1.70i·22-s − 0.487·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.34i·5-s + (−0.269 + 0.962i)7-s − 0.353i·8-s + 0.947·10-s − 0.512·11-s + 0.461·13-s + (−0.680 − 0.190i)14-s + 0.250·16-s − 1.08i·17-s + (0.490 + 0.871i)19-s + 0.670i·20-s − 0.362i·22-s − 0.101·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9674552682\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9674552682\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.713 - 2.54i)T \) |
| 19 | \( 1 + (-2.13 - 3.79i)T \) |
good | 5 | \( 1 + 2.99iT - 5T^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 - 1.66T + 13T^{2} \) |
| 17 | \( 1 + 4.46iT - 17T^{2} \) |
| 23 | \( 1 + 0.487T + 23T^{2} \) |
| 29 | \( 1 + 1.42iT - 29T^{2} \) |
| 31 | \( 1 + 3.18T + 31T^{2} \) |
| 37 | \( 1 + 9.68iT - 37T^{2} \) |
| 41 | \( 1 - 3.63T + 41T^{2} \) |
| 43 | \( 1 + 4.40T + 43T^{2} \) |
| 47 | \( 1 - 2.80iT - 47T^{2} \) |
| 53 | \( 1 + 8.53iT - 53T^{2} \) |
| 59 | \( 1 + 9.05T + 59T^{2} \) |
| 61 | \( 1 - 3.96iT - 61T^{2} \) |
| 67 | \( 1 - 2.67iT - 67T^{2} \) |
| 71 | \( 1 + 8.53iT - 71T^{2} \) |
| 73 | \( 1 + 3.63iT - 73T^{2} \) |
| 79 | \( 1 + 14.7iT - 79T^{2} \) |
| 83 | \( 1 - 1.60iT - 83T^{2} \) |
| 89 | \( 1 - 7.27T + 89T^{2} \) |
| 97 | \( 1 + 4.57T + 97T^{2} \) |
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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
−8.847710954927794389102273640980, −8.050950170887110828335061911939, −7.44888583456774185968162445559, −6.29996577672888075996886210353, −5.53882683029252283092517678839, −5.13959508961647100556851298196, −4.19587971347236795127800054987, −3.08305657370018756735867286886, −1.75382875123150865279829929701, −0.34555071055414300820388454701,
1.27933890130997330738388694124, 2.58193088139731441821124457627, 3.31569296357258093683837451301, 3.99666559985468062079882989856, 5.04055970781736842525549727517, 6.17759598242250016733598401834, 6.81159145556131031277644748469, 7.58360574943968446857974570426, 8.326642005119157511759195000636, 9.370381829877402280576175910776