L(s) = 1 | − 0.586i·3-s + 5-s + (2.52 − 0.792i)7-s + 2.65·9-s + 0.580·11-s + 1.14·13-s − 0.586i·15-s + 1.82i·17-s + 4.72i·19-s + (−0.464 − 1.48i)21-s + 2.79i·23-s + 25-s − 3.31i·27-s − 7.40i·29-s − 4.73·31-s + ⋯ |
L(s) = 1 | − 0.338i·3-s + 0.447·5-s + (0.954 − 0.299i)7-s + 0.885·9-s + 0.174·11-s + 0.318·13-s − 0.151i·15-s + 0.443i·17-s + 1.08i·19-s + (−0.101 − 0.323i)21-s + 0.583i·23-s + 0.200·25-s − 0.638i·27-s − 1.37i·29-s − 0.849·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.159996461\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.159996461\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (-2.52 + 0.792i)T \) |
good | 3 | \( 1 + 0.586iT - 3T^{2} \) |
| 11 | \( 1 - 0.580T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 - 1.82iT - 17T^{2} \) |
| 19 | \( 1 - 4.72iT - 19T^{2} \) |
| 23 | \( 1 - 2.79iT - 23T^{2} \) |
| 29 | \( 1 + 7.40iT - 29T^{2} \) |
| 31 | \( 1 + 4.73T + 31T^{2} \) |
| 37 | \( 1 + 4.35iT - 37T^{2} \) |
| 41 | \( 1 - 8.46iT - 41T^{2} \) |
| 43 | \( 1 - 4.32T + 43T^{2} \) |
| 47 | \( 1 + 3.56T + 47T^{2} \) |
| 53 | \( 1 + 5.48iT - 53T^{2} \) |
| 59 | \( 1 - 13.2iT - 59T^{2} \) |
| 61 | \( 1 - 0.275T + 61T^{2} \) |
| 67 | \( 1 - 7.71T + 67T^{2} \) |
| 71 | \( 1 + 1.13iT - 71T^{2} \) |
| 73 | \( 1 + 1.78iT - 73T^{2} \) |
| 79 | \( 1 + 10.0iT - 79T^{2} \) |
| 83 | \( 1 + 15.6iT - 83T^{2} \) |
| 89 | \( 1 + 15.3iT - 89T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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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
−9.898219234668103251416131744532, −8.962070324065861960557710109861, −7.937739946138428539497302577198, −7.49331661184495996184209636628, −6.39183153436944914629485212698, −5.63300752077331534882352139749, −4.51118359030741562879624557075, −3.72058393773443165838175475549, −2.06855794261356626166623213206, −1.27812303395560571428903490433,
1.29057320740195518234470933999, 2.43657833838884936361749069004, 3.78978676491489241728180041432, 4.82198819534013683574682193926, 5.35867789594014731387511337437, 6.64260369969837795495041024119, 7.29389868933906442326016467000, 8.393503264823850322951505761027, 9.102202314118318878410892637889, 9.788538708972076041749958401674