L(s) = 1 | + (0.5 + 0.866i)2-s + (0.965 + 0.258i)3-s + (−0.499 + 0.866i)4-s + (−0.0421 − 2.23i)5-s + (0.258 + 0.965i)6-s + (2.21 + 1.27i)7-s − 0.999·8-s + (0.866 + 0.499i)9-s + (1.91 − 1.15i)10-s + (0.240 − 0.897i)11-s + (−0.707 + 0.707i)12-s + (3.05 + 1.91i)13-s + 2.55i·14-s + (0.537 − 2.17i)15-s + (−0.5 − 0.866i)16-s + (0.495 + 1.84i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.557 + 0.149i)3-s + (−0.249 + 0.433i)4-s + (−0.0188 − 0.999i)5-s + (0.105 + 0.394i)6-s + (0.836 + 0.482i)7-s − 0.353·8-s + (0.288 + 0.166i)9-s + (0.605 − 0.365i)10-s + (0.0725 − 0.270i)11-s + (−0.204 + 0.204i)12-s + (0.846 + 0.531i)13-s + 0.682i·14-s + (0.138 − 0.560i)15-s + (−0.125 − 0.216i)16-s + (0.120 + 0.448i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.90067 + 0.771726i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.90067 + 0.771726i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.965 - 0.258i)T \) |
| 5 | \( 1 + (0.0421 + 2.23i)T \) |
| 13 | \( 1 + (-3.05 - 1.91i)T \) |
good | 7 | \( 1 + (-2.21 - 1.27i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.240 + 0.897i)T + (-9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-0.495 - 1.84i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.321 + 0.0861i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.516 + 1.92i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-1.58 + 0.914i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.423 - 0.423i)T - 31iT^{2} \) |
| 37 | \( 1 + (6.15 - 3.55i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (10.7 + 2.88i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (1.38 - 0.371i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + 7.62iT - 47T^{2} \) |
| 53 | \( 1 + (0.567 - 0.567i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.37 - 5.13i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (5.39 - 9.34i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.90 + 11.9i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.55 + 13.2i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 - 10.6iT - 79T^{2} \) |
| 83 | \( 1 + 6.12iT - 83T^{2} \) |
| 89 | \( 1 + (-6.90 - 1.85i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-1.12 + 1.94i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.71136907220048970198575008680, −10.46564383388899287784330419256, −9.143318383782099234469598966784, −8.551709829195321177824021389846, −8.008114719516992174539029076540, −6.64373851936446305361071411917, −5.47044129453414461374350788369, −4.64186430269176918271803124529, −3.56871661181340014438200716509, −1.73403590585548094611021206087,
1.59649887871542475554794683446, 2.98276332564841662842557725520, 3.89397034035204707959000158464, 5.18101943612647775256695161138, 6.51303182518641438679241339873, 7.50724760989011900794152314380, 8.387982735128239904750484006590, 9.595692123146066909228501192751, 10.49643810082172586520854459768, 11.12958823503459518015166851460