L(s) = 1 | + (0.587 − 1.80i)2-s + (0.809 − 0.587i)3-s + (−2.11 − 1.53i)4-s + (0.363 + 1.11i)5-s + (−0.587 − 1.80i)6-s + (−2.48 + 1.80i)8-s + (0.309 − 0.951i)9-s + 2.23·10-s + (−0.951 + 0.309i)11-s − 2.61·12-s + (−0.309 + 0.951i)13-s + (0.951 + 0.690i)15-s + (1.00 + 3.07i)16-s + (−1.53 − 1.11i)18-s + (0.951 − 2.92i)20-s + ⋯ |
L(s) = 1 | + (0.587 − 1.80i)2-s + (0.809 − 0.587i)3-s + (−2.11 − 1.53i)4-s + (0.363 + 1.11i)5-s + (−0.587 − 1.80i)6-s + (−2.48 + 1.80i)8-s + (0.309 − 0.951i)9-s + 2.23·10-s + (−0.951 + 0.309i)11-s − 2.61·12-s + (−0.309 + 0.951i)13-s + (0.951 + 0.690i)15-s + (1.00 + 3.07i)16-s + (−1.53 − 1.11i)18-s + (0.951 − 2.92i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.233695733\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.233695733\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (0.951 - 0.309i)T \) |
| 13 | \( 1 + (0.309 - 0.951i)T \) |
good | 2 | \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.809 + 0.587i)T^{2} \) |
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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
−11.07112914635642544512596210746, −10.25539082679420940330141780588, −9.625063661463710144896538820845, −8.671179532834960810511145191449, −7.33666265734262180258665168598, −6.19476176494610659380490603999, −4.78666277226373006225493738430, −3.55073369425341405866554275408, −2.64677487073326551684795938580, −1.90755163675949141804022711470,
3.06899231554797084728625985944, 4.38133279910081686622752697949, 5.15878342154403649132149615941, 5.77203694809446698676076164616, 7.29941588115205561415771861110, 8.080110184418893923874602965934, 8.660211528086965297391715130500, 9.446398202025009898992147011060, 10.45189091315671496983927600678, 12.32426096729215313894333303025