| L(s) = 1 | + (0.258 − 0.965i)3-s + (−1.93 − 1.12i)5-s + (−1.60 + 2.10i)7-s + (−0.866 − 0.499i)9-s + (2.59 + 4.49i)11-s + (0.393 + 0.393i)13-s + (−1.58 + 1.57i)15-s + (4.10 + 1.10i)17-s + (2.32 − 4.02i)19-s + (1.62 + 2.09i)21-s + (0.979 + 3.65i)23-s + (2.48 + 4.33i)25-s + (−0.707 + 0.707i)27-s + 1.14i·29-s + (9.03 − 5.21i)31-s + ⋯ |
| L(s) = 1 | + (0.149 − 0.557i)3-s + (−0.865 − 0.501i)5-s + (−0.604 + 0.796i)7-s + (−0.288 − 0.166i)9-s + (0.781 + 1.35i)11-s + (0.109 + 0.109i)13-s + (−0.408 + 0.407i)15-s + (0.996 + 0.267i)17-s + (0.533 − 0.923i)19-s + (0.353 + 0.456i)21-s + (0.204 + 0.762i)23-s + (0.497 + 0.867i)25-s + (−0.136 + 0.136i)27-s + 0.211i·29-s + (1.62 − 0.937i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29752 + 0.195915i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29752 + 0.195915i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 + (1.93 + 1.12i)T \) |
| 7 | \( 1 + (1.60 - 2.10i)T \) |
| good | 11 | \( 1 + (-2.59 - 4.49i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.393 - 0.393i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.10 - 1.10i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-2.32 + 4.02i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.979 - 3.65i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 - 1.14iT - 29T^{2} \) |
| 31 | \( 1 + (-9.03 + 5.21i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.54 - 1.75i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 3.89iT - 41T^{2} \) |
| 43 | \( 1 + (-8.49 + 8.49i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.37 - 8.87i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (3.74 + 1.00i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-1.64 - 2.85i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.21 - 1.85i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.26 - 8.44i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + (2.84 - 10.6i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-14.8 - 8.57i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.465 - 0.465i)T + 83iT^{2} \) |
| 89 | \( 1 + (-1.44 + 2.49i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.58 + 3.58i)T - 97iT^{2} \) |
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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
−10.00405232655235667023293938259, −9.292911063871063259776756202604, −8.601658251390555936066520662349, −7.57501347189121063884734317933, −6.99721247040464221325492889917, −5.93127649395940981837031424294, −4.86808791677303635283814399693, −3.81043338249327075321958029503, −2.67594843700333457625787758631, −1.20663070841444412157755894503,
0.77693343416722606095015029268, 3.17465826786129322980070476885, 3.50759196527229399380625107717, 4.53888114358075864289328584194, 5.89200866173129842658611329691, 6.69802587906932871808452448511, 7.70203401548314704445027860295, 8.420710978716986583090624980178, 9.362144289180160721358721471124, 10.39608533250908969719820091335
