| L(s) = 1 | + (−0.959 + 0.281i)2-s + (−1.73 + 1.50i)3-s + (0.841 − 0.540i)4-s + (0.406 + 2.82i)5-s + (1.24 − 1.93i)6-s + (0.697 + 2.55i)7-s + (−0.654 + 0.755i)8-s + (0.326 − 2.27i)9-s + (−1.18 − 2.59i)10-s + (−0.284 + 0.970i)11-s + (−0.648 + 2.20i)12-s + (3.12 − 1.42i)13-s + (−1.38 − 2.25i)14-s + (−4.96 − 4.30i)15-s + (0.415 − 0.909i)16-s + (0.250 + 0.160i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (−1.00 + 0.870i)3-s + (0.420 − 0.270i)4-s + (0.181 + 1.26i)5-s + (0.507 − 0.790i)6-s + (0.263 + 0.964i)7-s + (−0.231 + 0.267i)8-s + (0.108 − 0.757i)9-s + (−0.375 − 0.821i)10-s + (−0.0859 + 0.292i)11-s + (−0.187 + 0.637i)12-s + (0.865 − 0.395i)13-s + (−0.371 − 0.601i)14-s + (−1.28 − 1.11i)15-s + (0.103 − 0.227i)16-s + (0.0607 + 0.0390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0177708 + 0.601469i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0177708 + 0.601469i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 7 | \( 1 + (-0.697 - 2.55i)T \) |
| 23 | \( 1 + (3.22 - 3.55i)T \) |
| good | 3 | \( 1 + (1.73 - 1.50i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-0.406 - 2.82i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (0.284 - 0.970i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-3.12 + 1.42i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.250 - 0.160i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (2.14 - 1.37i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (4.42 + 2.84i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (4.54 + 3.93i)T + (4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (-3.13 - 0.450i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (1.66 - 0.239i)T + (39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-8.73 + 7.57i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 7.57iT - 47T^{2} \) |
| 53 | \( 1 + (-7.98 - 3.64i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-6.61 + 3.02i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-6.57 + 7.59i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-4.29 - 14.6i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (1.97 - 0.581i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-1.61 - 2.51i)T + (-30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (15.2 - 6.95i)T + (51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (1.06 - 7.39i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-0.142 - 0.164i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (2.49 + 17.3i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.40802600851034273456311448508, −11.17567085434423807376204126187, −10.24223415599996814767372306430, −9.593697636560849984080819075354, −8.378992863780224328605315386007, −7.23871429815221149286321994022, −5.94264526451139805324191211573, −5.65527266477819216571976931200, −3.93152982602878592100611035918, −2.31839602462568707850270006814,
0.62567593133452750862119857108, 1.63527908939026703455351478585, 4.03602297655365593128665225350, 5.34660343525159030275069227436, 6.39848217578458624131189655164, 7.32068364535650243780725841737, 8.376920260961008880581785472633, 9.155960527810556315570262091097, 10.47116023170274415627054912336, 11.18713646356045499082055094685
