| L(s) = 1 | + (−1.35 − 0.410i)2-s + (−1.57 − 1.96i)3-s + (1.66 + 1.11i)4-s + (1.55 − 1.23i)5-s + (1.31 + 3.31i)6-s + (2.03 − 1.69i)7-s + (−1.79 − 2.18i)8-s + (−0.745 + 3.26i)9-s + (−2.61 + 1.03i)10-s + (3.09 − 0.707i)11-s + (−0.423 − 5.02i)12-s + (−5.38 + 1.22i)13-s + (−3.44 + 1.45i)14-s + (−4.88 − 1.11i)15-s + (1.53 + 3.69i)16-s + (2.48 − 5.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.956 − 0.290i)2-s + (−0.906 − 1.13i)3-s + (0.831 + 0.555i)4-s + (0.695 − 0.554i)5-s + (0.537 + 1.35i)6-s + (0.768 − 0.639i)7-s + (−0.634 − 0.772i)8-s + (−0.248 + 1.08i)9-s + (−0.826 + 0.328i)10-s + (0.934 − 0.213i)11-s + (−0.122 − 1.44i)12-s + (−1.49 + 0.340i)13-s + (−0.921 + 0.388i)14-s + (−1.26 − 0.287i)15-s + (0.382 + 0.923i)16-s + (0.601 − 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.241972 - 0.600233i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.241972 - 0.600233i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.35 + 0.410i)T \) |
| 7 | \( 1 + (-2.03 + 1.69i)T \) |
| good | 3 | \( 1 + (1.57 + 1.96i)T + (-0.667 + 2.92i)T^{2} \) |
| 5 | \( 1 + (-1.55 + 1.23i)T + (1.11 - 4.87i)T^{2} \) |
| 11 | \( 1 + (-3.09 + 0.707i)T + (9.91 - 4.77i)T^{2} \) |
| 13 | \( 1 + (5.38 - 1.22i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-2.48 + 5.15i)T + (-10.5 - 13.2i)T^{2} \) |
| 19 | \( 1 + 7.59T + 19T^{2} \) |
| 23 | \( 1 + (0.751 + 1.55i)T + (-14.3 + 17.9i)T^{2} \) |
| 29 | \( 1 + (-0.626 - 0.301i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 - 8.05T + 31T^{2} \) |
| 37 | \( 1 + (-4.24 - 2.04i)T + (23.0 + 28.9i)T^{2} \) |
| 41 | \( 1 + (0.0970 - 0.0773i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (-3.53 - 2.81i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (0.299 + 1.31i)T + (-42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-0.183 + 0.0884i)T + (33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + (1.40 - 1.76i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (-0.524 + 1.08i)T + (-38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + 3.71iT - 67T^{2} \) |
| 71 | \( 1 + (3.42 + 7.11i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-4.61 - 1.05i)T + (65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 - 9.99iT - 79T^{2} \) |
| 83 | \( 1 + (-0.712 + 3.12i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (-14.8 - 3.38i)T + (80.1 + 38.6i)T^{2} \) |
| 97 | \( 1 + 3.30iT - 97T^{2} \) |
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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
−11.97585010062639679658597880447, −11.34338368798777330622395574069, −10.17424725409516045080507536133, −9.203481197465715730878133652958, −7.959737762252955192199130765441, −7.04860162187499576956628016672, −6.21964966996212499539649870352, −4.71066106995781362659516513879, −2.12514264263622905200503138540, −0.882813444233620004289886396439,
2.21229052647269112522790722709, 4.51207884083504168146727279472, 5.74015221537554484640701029819, 6.43826795972425448937625474607, 7.978070599038829003988284462049, 9.155360396021029820823082270524, 10.13358827666362227220979468298, 10.50031911030801684264612671662, 11.57773595532288350880826193071, 12.34825800178513904051970835048
