L(s) = 1 | + 2i·2-s − 6.79i·3-s − 4·4-s + (−10.7 + 2.94i)5-s + 13.5·6-s − 8i·8-s − 19.1·9-s + (−5.88 − 21.5i)10-s + 69.2·11-s + 27.1i·12-s + 28.3i·13-s + (19.9 + 73.2i)15-s + 16·16-s + 11.3i·17-s − 38.2i·18-s − 74.2·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.30i·3-s − 0.5·4-s + (−0.964 + 0.263i)5-s + 0.924·6-s − 0.353i·8-s − 0.708·9-s + (−0.186 − 0.682i)10-s + 1.89·11-s + 0.653i·12-s + 0.605i·13-s + (0.344 + 1.26i)15-s + 0.250·16-s + 0.161i·17-s − 0.500i·18-s − 0.896·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.334848610\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.334848610\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2iT \) |
| 5 | \( 1 + (10.7 - 2.94i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 6.79iT - 27T^{2} \) |
| 11 | \( 1 - 69.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 28.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 11.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 74.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 66.6iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 160.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 284. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 149.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 513. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 411. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 354. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 210.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 587.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.03e3iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 743. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 609.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.14e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 735.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 333. iT - 9.12e5T^{2} \) |
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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
−10.40357264440817623053843848378, −8.971212612477930470693716831967, −8.437363601160836320642022374284, −7.39121923292156193274435114010, −6.70623915187515123407886979103, −6.29787802928317776253613224563, −4.54716257531683323064130783183, −3.63756393917507925443023124969, −1.82904049002007289691857292996, −0.50919014212979268343927276855,
1.14355927410256486757505670828, 3.10781639775541402314470599337, 4.09858867679632129573662923088, 4.41299457919885260736609439625, 5.77628846482881612514318294144, 7.19103669117804412003727752199, 8.461522632866984044238311523360, 9.160449340949245463534826451844, 9.812915323479232639964212433072, 10.92230662656317878347505877462