| L(s) = 1 | + (1.31 − 0.524i)2-s + (0.349 + 1.75i)3-s + (1.44 − 1.37i)4-s + (−1.08 + 1.62i)5-s + (1.38 + 2.12i)6-s + (−0.680 − 1.01i)7-s + (1.18 − 2.57i)8-s + (−0.199 + 0.0825i)9-s + (−0.574 + 2.70i)10-s + (−3.15 − 0.628i)11-s + (2.93 + 2.06i)12-s + (0.472 + 0.472i)13-s + (−1.42 − 0.980i)14-s + (−3.24 − 1.34i)15-s + (0.201 − 3.99i)16-s + (0.972 − 4.00i)17-s + ⋯ |
| L(s) = 1 | + (0.928 − 0.371i)2-s + (0.201 + 1.01i)3-s + (0.724 − 0.689i)4-s + (−0.486 + 0.727i)5-s + (0.564 + 0.868i)6-s + (−0.257 − 0.385i)7-s + (0.417 − 0.908i)8-s + (−0.0664 + 0.0275i)9-s + (−0.181 + 0.856i)10-s + (−0.952 − 0.189i)11-s + (0.846 + 0.596i)12-s + (0.131 + 0.131i)13-s + (−0.381 − 0.262i)14-s + (−0.837 − 0.346i)15-s + (0.0502 − 0.998i)16-s + (0.235 − 0.971i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70669 + 0.220162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70669 + 0.220162i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.31 + 0.524i)T \) |
| 17 | \( 1 + (-0.972 + 4.00i)T \) |
| good | 3 | \( 1 + (-0.349 - 1.75i)T + (-2.77 + 1.14i)T^{2} \) |
| 5 | \( 1 + (1.08 - 1.62i)T + (-1.91 - 4.61i)T^{2} \) |
| 7 | \( 1 + (0.680 + 1.01i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (3.15 + 0.628i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-0.472 - 0.472i)T + 13iT^{2} \) |
| 19 | \( 1 + (0.347 + 0.143i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (7.81 + 1.55i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.263 - 0.176i)T + (11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (-0.751 - 3.77i)T + (-28.6 + 11.8i)T^{2} \) |
| 37 | \( 1 + (-6.83 + 1.35i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (5.01 - 3.35i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (7.39 - 3.06i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-6.28 - 6.28i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.69 - 11.3i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (1.04 + 2.52i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-9.24 + 6.17i)T + (23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 - 1.57iT - 67T^{2} \) |
| 71 | \( 1 + (-9.54 + 1.89i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-3.44 - 2.30i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-2.67 + 13.4i)T + (-72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-2.64 + 6.38i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-11.0 - 11.0i)T + 89iT^{2} \) |
| 97 | \( 1 + (2.39 - 3.58i)T + (-37.1 - 89.6i)T^{2} \) |
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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
−13.40461108525635341425465465168, −12.21044390190534517093855768752, −11.10610487879785566233515953750, −10.39005802969415190827671864436, −9.599534712685860261139652084403, −7.75095564531728150475199587978, −6.55634913382672374577581682136, −5.05168165591106301210932354439, −3.91940469366351715155240682870, −2.91365678354877452919173710166,
2.21091315716842910152576995275, 4.01509270444521640824402315264, 5.43896089844346232434089691891, 6.57807672673842056919288548075, 7.897959009843427420254776188147, 8.278698797708340775126985137479, 10.23283954608603497206652475605, 11.76467099714992948295380703262, 12.48706590608513800631794018243, 13.06376100335000039035273319228
