L(s) = 1 | + (1.85 + 0.739i)2-s − 0.0974·3-s + (2.90 + 2.74i)4-s + 3.46i·5-s + (−0.181 − 0.0720i)6-s + (3.37 + 7.25i)8-s − 8.99·9-s + (−2.56 + 6.44i)10-s − 2.92·11-s + (−0.283 − 0.267i)12-s + 19.1i·13-s − 0.337i·15-s + (0.902 + 15.9i)16-s + 14.3·17-s + (−16.7 − 6.64i)18-s − 8.09·19-s + ⋯ |
L(s) = 1 | + (0.929 + 0.369i)2-s − 0.0324·3-s + (0.726 + 0.686i)4-s + 0.693i·5-s + (−0.0301 − 0.0120i)6-s + (0.421 + 0.906i)8-s − 0.998·9-s + (−0.256 + 0.644i)10-s − 0.266·11-s + (−0.0236 − 0.0223i)12-s + 1.47i·13-s − 0.0225i·15-s + (0.0563 + 0.998i)16-s + 0.846·17-s + (−0.928 − 0.369i)18-s − 0.426·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.421 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.37234 + 2.15103i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37234 + 2.15103i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.85 - 0.739i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 0.0974T + 9T^{2} \) |
| 5 | \( 1 - 3.46iT - 25T^{2} \) |
| 11 | \( 1 + 2.92T + 121T^{2} \) |
| 13 | \( 1 - 19.1iT - 169T^{2} \) |
| 17 | \( 1 - 14.3T + 289T^{2} \) |
| 19 | \( 1 + 8.09T + 361T^{2} \) |
| 23 | \( 1 - 16.7iT - 529T^{2} \) |
| 29 | \( 1 + 27.1iT - 841T^{2} \) |
| 31 | \( 1 + 44.8iT - 961T^{2} \) |
| 37 | \( 1 - 39.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 45.8T + 1.68e3T^{2} \) |
| 43 | \( 1 - 61.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 46.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 9.69iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 114.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 7.48iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 12.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 129. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 18.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 42.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 109.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 80.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 162.T + 9.40e3T^{2} \) |
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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
−11.53445002463004371855037072662, −10.83868315600956579795806217713, −9.568715878313498134958999293918, −8.366804776560235120313391785865, −7.44433830845108774146082660949, −6.46200170978775108731019993920, −5.72373772628740564626380721620, −4.48923716203443472841476439037, −3.33307115006048643328231826525, −2.24963277963283552312940691819,
0.825190252622306324028297897711, 2.61083633661196776965423302115, 3.62601473433150878941801920387, 5.17685805962019710333999574211, 5.47286898016566660443994461315, 6.78218889115875896040297133965, 8.062179708874485100909467222631, 8.922589028994573763341690917330, 10.31801399539459177396171381314, 10.77449633025070290034977709095