L(s) = 1 | + (−0.891 − 0.453i)2-s + (0.587 + 0.809i)4-s + (0.453 − 0.891i)5-s + (0.221 + 0.221i)7-s + (−0.156 − 0.987i)8-s + (−0.809 + 0.587i)10-s + (1.69 − 0.550i)11-s + (−0.0966 − 0.297i)14-s + (−0.309 + 0.951i)16-s + (0.987 − 0.156i)20-s + (−1.76 − 0.278i)22-s + (−0.587 − 0.809i)25-s + (−0.0489 + 0.309i)28-s + (−1.14 + 0.831i)29-s + (0.951 + 0.690i)31-s + (0.707 − 0.707i)32-s + ⋯ |
L(s) = 1 | + (−0.891 − 0.453i)2-s + (0.587 + 0.809i)4-s + (0.453 − 0.891i)5-s + (0.221 + 0.221i)7-s + (−0.156 − 0.987i)8-s + (−0.809 + 0.587i)10-s + (1.69 − 0.550i)11-s + (−0.0966 − 0.297i)14-s + (−0.309 + 0.951i)16-s + (0.987 − 0.156i)20-s + (−1.76 − 0.278i)22-s + (−0.587 − 0.809i)25-s + (−0.0489 + 0.309i)28-s + (−1.14 + 0.831i)29-s + (0.951 + 0.690i)31-s + (0.707 − 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9240023702\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9240023702\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.891 + 0.453i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.453 + 0.891i)T \) |
good | 7 | \( 1 + (-0.221 - 0.221i)T + iT^{2} \) |
| 11 | \( 1 + (-1.69 + 0.550i)T + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (0.587 + 0.809i)T^{2} \) |
| 29 | \( 1 + (1.14 - 0.831i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 53 | \( 1 + (1.87 + 0.297i)T + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (-0.280 + 0.863i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-1.26 - 0.642i)T + (0.587 + 0.809i)T^{2} \) |
| 79 | \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.253 - 1.59i)T + (-0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (1.95 + 0.309i)T + (0.951 + 0.309i)T^{2} \) |
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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
−9.504990364082055315200980567072, −8.545980678674553042408568560896, −8.253896174538416370307939733661, −6.96645779336774961009963826814, −6.34226822715421083273614053354, −5.31459078468108483628560729626, −4.18739767540167471162865616266, −3.31890503797611067614947536596, −1.93462041303270532302091887443, −1.11806192537961712064526067164,
1.41915074921526641482912272671, 2.36837275251692052400034120012, 3.69283784198563667167026959498, 4.79447054444231995168578327327, 6.15206837891954116068232086305, 6.33853881068192709323469238207, 7.33263686393634507646680891684, 7.83630901441613944920036555093, 9.037143612648083744026527544627, 9.498408037276563698687029918041