L(s) = 1 | + (0.881 − 0.471i)2-s + (0.634 − 0.773i)3-s + (0.555 − 0.831i)4-s + (−0.956 + 0.290i)5-s + (0.195 − 0.980i)6-s + (0.0980 − 0.995i)8-s + (−0.195 − 0.980i)9-s + (−0.707 + 0.707i)10-s + (−0.290 − 0.956i)12-s + (−0.382 + 0.923i)15-s + (−0.382 − 0.923i)16-s + (−0.222 − 0.536i)17-s + (−0.634 − 0.773i)18-s + (−0.172 + 0.0924i)19-s + (−0.290 + 0.956i)20-s + ⋯ |
L(s) = 1 | + (0.881 − 0.471i)2-s + (0.634 − 0.773i)3-s + (0.555 − 0.831i)4-s + (−0.956 + 0.290i)5-s + (0.195 − 0.980i)6-s + (0.0980 − 0.995i)8-s + (−0.195 − 0.980i)9-s + (−0.707 + 0.707i)10-s + (−0.290 − 0.956i)12-s + (−0.382 + 0.923i)15-s + (−0.382 − 0.923i)16-s + (−0.222 − 0.536i)17-s + (−0.634 − 0.773i)18-s + (−0.172 + 0.0924i)19-s + (−0.290 + 0.956i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.427 + 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.960638881\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.960638881\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.881 + 0.471i)T \) |
| 3 | \( 1 + (-0.634 + 0.773i)T \) |
| 5 | \( 1 + (0.956 - 0.290i)T \) |
good | 7 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 11 | \( 1 + (-0.980 - 0.195i)T^{2} \) |
| 13 | \( 1 + (0.831 + 0.555i)T^{2} \) |
| 17 | \( 1 + (0.222 + 0.536i)T + (-0.707 + 0.707i)T^{2} \) |
| 19 | \( 1 + (0.172 - 0.0924i)T + (0.555 - 0.831i)T^{2} \) |
| 23 | \( 1 + (-0.523 + 0.783i)T + (-0.382 - 0.923i)T^{2} \) |
| 29 | \( 1 + (0.980 - 0.195i)T^{2} \) |
| 31 | \( 1 + (-0.785 - 0.785i)T + iT^{2} \) |
| 37 | \( 1 + (0.555 + 0.831i)T^{2} \) |
| 41 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 43 | \( 1 + (0.195 - 0.980i)T^{2} \) |
| 47 | \( 1 + (1.42 - 0.591i)T + (0.707 - 0.707i)T^{2} \) |
| 53 | \( 1 + (0.192 - 1.95i)T + (-0.980 - 0.195i)T^{2} \) |
| 59 | \( 1 + (0.831 - 0.555i)T^{2} \) |
| 61 | \( 1 + (-1.53 - 1.26i)T + (0.195 + 0.980i)T^{2} \) |
| 67 | \( 1 + (-0.195 - 0.980i)T^{2} \) |
| 71 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 73 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 79 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 + (-0.183 - 0.344i)T + (-0.555 + 0.831i)T^{2} \) |
| 89 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.049570262925818643163804892602, −8.294837768039089330646430908489, −7.36733834974651351273795888323, −6.82913926272804707714367368487, −6.07098172234225181114351260010, −4.83787888448688841311601961399, −4.05270984771955160166463664879, −3.09871709978350395090768624623, −2.49606601466182245441354291333, −1.06404152009707658671986968184,
2.15345502935193118722675993692, 3.35345736322554871177790952260, 3.83377309387879578893568216576, 4.72932344139347963430919694708, 5.30333958979020567012353862595, 6.48471510359340520853681402912, 7.36238680272867412745619586185, 8.150525020009614056764504839586, 8.535665238159769656888778969616, 9.502672879151392750205477940425