L(s) = 1 | + (0.358 + 0.207i)3-s + (−2.09 + 1.62i)7-s + (−1.41 − 2.44i)9-s + (−0.414 + 0.717i)11-s − 2i·13-s + (6.63 + 3.82i)17-s + (−2.82 − 4.89i)19-s + (−1.08 + 0.148i)21-s + (4.83 − 2.79i)23-s − 2.41i·27-s + 7.82·29-s + (−0.414 + 0.717i)31-s + (−0.297 + 0.171i)33-s + (4.89 − 2.82i)37-s + (0.414 − 0.717i)39-s + ⋯ |
L(s) = 1 | + (0.207 + 0.119i)3-s + (−0.790 + 0.612i)7-s + (−0.471 − 0.816i)9-s + (−0.124 + 0.216i)11-s − 0.554i·13-s + (1.60 + 0.928i)17-s + (−0.648 − 1.12i)19-s + (−0.236 + 0.0324i)21-s + (1.00 − 0.582i)23-s − 0.464i·27-s + 1.45·29-s + (−0.0743 + 0.128i)31-s + (−0.0517 + 0.0298i)33-s + (0.805 − 0.464i)37-s + (0.0663 − 0.114i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.564866556\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.564866556\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.09 - 1.62i)T \) |
good | 3 | \( 1 + (-0.358 - 0.207i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (0.414 - 0.717i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + (-6.63 - 3.82i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.82 + 4.89i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.83 + 2.79i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.82T + 29T^{2} \) |
| 31 | \( 1 + (0.414 - 0.717i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.89 + 2.82i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5.82T + 41T^{2} \) |
| 43 | \( 1 - 6.89iT - 43T^{2} \) |
| 47 | \( 1 + (-10.0 + 5.82i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.89 + 2.82i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.32 + 5.76i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.1 - 6.44i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-3.16 - 1.82i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.75iT - 83T^{2} \) |
| 89 | \( 1 + (2.67 + 4.62i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6iT - 97T^{2} \) |
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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
−9.465333010182237803671656093422, −8.758985411678033994720462923150, −8.091664619601178868742232395666, −6.96143947699934824547318614111, −6.18225432035764762138680587310, −5.51096826311665397055340853838, −4.33600902521602101460603718778, −3.19678711673711918065649894028, −2.64420714062935690305849652945, −0.77667698561954778086707424390,
1.07091295373138915143719320760, 2.61534675489169185368230019054, 3.40438575837282232481371456678, 4.50514150338525675111339908422, 5.52449324332511089162904332679, 6.31832882408389615005374473412, 7.38098771567315304857327699747, 7.83208615699703979326234610689, 8.842400896287619845900583420544, 9.643460549396677899421614858085