L(s) = 1 | + (0.104 + 0.994i)3-s + (0.913 − 0.406i)4-s + (−0.360 + 1.69i)5-s + (−0.978 + 0.207i)9-s + (0.499 + 0.866i)12-s + (−1.72 − 0.181i)15-s + (0.669 − 0.743i)16-s + (0.360 + 1.69i)20-s + (−1.82 − 0.813i)25-s + (−0.309 − 0.951i)27-s + (0.669 + 0.743i)31-s + (−0.809 + 0.587i)36-s + (−0.809 − 0.587i)37-s − 1.73i·45-s + (0.704 − 1.58i)47-s + (0.809 + 0.587i)48-s + ⋯ |
L(s) = 1 | + (0.104 + 0.994i)3-s + (0.913 − 0.406i)4-s + (−0.360 + 1.69i)5-s + (−0.978 + 0.207i)9-s + (0.499 + 0.866i)12-s + (−1.72 − 0.181i)15-s + (0.669 − 0.743i)16-s + (0.360 + 1.69i)20-s + (−1.82 − 0.813i)25-s + (−0.309 − 0.951i)27-s + (0.669 + 0.743i)31-s + (−0.809 + 0.587i)36-s + (−0.809 − 0.587i)37-s − 1.73i·45-s + (0.704 − 1.58i)47-s + (0.809 + 0.587i)48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.178559704\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.178559704\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 5 | \( 1 + (0.360 - 1.69i)T + (-0.913 - 0.406i)T^{2} \) |
| 7 | \( 1 + (-0.978 - 0.207i)T^{2} \) |
| 13 | \( 1 + (-0.104 + 0.994i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 31 | \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.704 + 1.58i)T + (-0.669 - 0.743i)T^{2} \) |
| 53 | \( 1 + (1.64 + 0.535i)T + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.704 - 1.58i)T + (-0.669 + 0.743i)T^{2} \) |
| 61 | \( 1 + (-0.104 - 0.994i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-1.64 + 0.535i)T + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.913 - 0.406i)T^{2} \) |
| 83 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \) |
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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
−10.40554290226392708733555438205, −9.873958060237264800573376805915, −8.705726974554076102402753410422, −7.63377732543839139104005855037, −6.90971257128013863199535622360, −6.16346037823480867905668232596, −5.25092539532557450184918084391, −3.88904984305880427926496917222, −3.08887070303505115743302534076, −2.29157425167734457438647947118,
1.12357714568535624351872737169, 2.19828096379616610277195692977, 3.47024417006653571864797481111, 4.67914736702576685346001322458, 5.72189369305370569249305708745, 6.53555945466509934014532738524, 7.56457887788090394294667547604, 8.085953218210742556771603173719, 8.732866919203795829433570102694, 9.621628007899242153387114560762