| L(s) = 1 | + (−1.41 − 0.0591i)2-s + (1.99 + 0.167i)4-s + 1.33i·7-s + (−2.80 − 0.353i)8-s − 2.94i·11-s − 2.04·13-s + (0.0788 − 1.88i)14-s + (3.94 + 0.665i)16-s − 3.61i·17-s − 5.35i·19-s + (−0.174 + 4.16i)22-s + 8.59i·23-s + (2.88 + 0.120i)26-s + (−0.222 + 2.65i)28-s + 5.26i·29-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0418i)2-s + (0.996 + 0.0835i)4-s + 0.504i·7-s + (−0.992 − 0.125i)8-s − 0.887i·11-s − 0.566·13-s + (0.0210 − 0.503i)14-s + (0.986 + 0.166i)16-s − 0.876i·17-s − 1.22i·19-s + (−0.0371 + 0.886i)22-s + 1.79i·23-s + (0.565 + 0.0236i)26-s + (−0.0421 + 0.502i)28-s + 0.977i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2092711381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2092711381\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 + 0.0591i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.33iT - 7T^{2} \) |
| 11 | \( 1 + 2.94iT - 11T^{2} \) |
| 13 | \( 1 + 2.04T + 13T^{2} \) |
| 17 | \( 1 + 3.61iT - 17T^{2} \) |
| 19 | \( 1 + 5.35iT - 19T^{2} \) |
| 23 | \( 1 - 8.59iT - 23T^{2} \) |
| 29 | \( 1 - 5.26iT - 29T^{2} \) |
| 31 | \( 1 + 2.08T + 31T^{2} \) |
| 37 | \( 1 + 6.55T + 37T^{2} \) |
| 41 | \( 1 + 7.02T + 41T^{2} \) |
| 43 | \( 1 + 8.50T + 43T^{2} \) |
| 47 | \( 1 + 9.97iT - 47T^{2} \) |
| 53 | \( 1 - 6.12T + 53T^{2} \) |
| 59 | \( 1 - 4.75iT - 59T^{2} \) |
| 61 | \( 1 + 8.51iT - 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 2.62T + 71T^{2} \) |
| 73 | \( 1 - 15.3iT - 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 1.52T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 + 13.4iT - 97T^{2} \) |
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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
−8.825436588948438950399670540071, −8.456729023585531232577226099968, −7.14191324767064544702079735366, −7.03747238381798664516107693133, −5.66730691894631336650128531092, −5.15285133448798822208501360274, −3.48766310682534179808763498455, −2.75737575083907913364031019014, −1.57617244926847376023931070120, −0.10508159558849959157051779854,
1.49689514959603011246890604643, 2.42369167222421464570706374563, 3.69442806114729665760363575190, 4.69289982276393742620380733323, 5.88592276061721855101433828250, 6.65727491365963151399163777848, 7.37763830802170255295522759940, 8.112506548087769817745087189997, 8.755608256323972070375686219258, 9.758162037570616013560545025059
