| L(s) = 1 | + 0.872i·2-s + 1.23·4-s + (1.62 + 1.53i)5-s + 4.38i·7-s + 2.82i·8-s + (−1.33 + 1.42i)10-s + 3.02·11-s − i·13-s − 3.82·14-s + 0.00847·16-s + 4.42i·17-s − 7.12·19-s + (2.01 + 1.89i)20-s + 2.64i·22-s − 5.41i·23-s + ⋯ |
| L(s) = 1 | + 0.617i·2-s + 0.618·4-s + (0.728 + 0.685i)5-s + 1.65i·7-s + 0.999i·8-s + (−0.422 + 0.449i)10-s + 0.911·11-s − 0.277i·13-s − 1.02·14-s + 0.00211·16-s + 1.07i·17-s − 1.63·19-s + (0.450 + 0.424i)20-s + 0.562i·22-s − 1.12i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.728 - 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.425129461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.425129461\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.62 - 1.53i)T \) |
| 13 | \( 1 + iT \) |
| good | 2 | \( 1 - 0.872iT - 2T^{2} \) |
| 7 | \( 1 - 4.38iT - 7T^{2} \) |
| 11 | \( 1 - 3.02T + 11T^{2} \) |
| 17 | \( 1 - 4.42iT - 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 + 5.41iT - 23T^{2} \) |
| 29 | \( 1 - 2.11T + 29T^{2} \) |
| 31 | \( 1 + 1.03T + 31T^{2} \) |
| 37 | \( 1 + 4.03iT - 37T^{2} \) |
| 41 | \( 1 - 7.11T + 41T^{2} \) |
| 43 | \( 1 + 6.80iT - 43T^{2} \) |
| 47 | \( 1 + 7.79iT - 47T^{2} \) |
| 53 | \( 1 + 7.78iT - 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 + 13.6iT - 67T^{2} \) |
| 71 | \( 1 - 8.32T + 71T^{2} \) |
| 73 | \( 1 - 0.0990iT - 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 6.68iT - 83T^{2} \) |
| 89 | \( 1 + 6.39T + 89T^{2} \) |
| 97 | \( 1 - 6.42iT - 97T^{2} \) |
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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
−9.390429871101572940565343491810, −8.681054747700420111041404190851, −8.152727322868568371714916873888, −6.90538466901407110614230629893, −6.26219862206016907480986924702, −5.98771330138850269653209731402, −5.03842105851577778000330324975, −3.61446047736376557813388941105, −2.31285153616852253861517374848, −2.06150750658700266660477638842,
0.900697868279223605782596761836, 1.63675842050999852592736637515, 2.85375672345641598086948908576, 4.05867926707569334471546406632, 4.51486114748403948719964510730, 5.91968305360726911098951724235, 6.68137831353877158042817386628, 7.25778793297455560403266380784, 8.228939400051897569933949879507, 9.447045078078007087441349724275
