L(s) = 1 | + (1.39 + 0.238i)2-s + (1.88 + 0.664i)4-s − i·5-s + (2.37 + 1.16i)7-s + (2.47 + 1.37i)8-s + (0.238 − 1.39i)10-s + 4.86i·11-s + 3.63i·13-s + (3.03 + 2.18i)14-s + (3.11 + 2.50i)16-s − 4.47i·17-s − 2.70·19-s + (0.664 − 1.88i)20-s + (−1.16 + 6.78i)22-s − 1.68i·23-s + ⋯ |
L(s) = 1 | + (0.985 + 0.168i)2-s + (0.943 + 0.332i)4-s − 0.447i·5-s + (0.898 + 0.439i)7-s + (0.873 + 0.486i)8-s + (0.0754 − 0.440i)10-s + 1.46i·11-s + 1.00i·13-s + (0.811 + 0.584i)14-s + (0.779 + 0.627i)16-s − 1.08i·17-s − 0.621·19-s + (0.148 − 0.421i)20-s + (−0.247 + 1.44i)22-s − 0.351i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.452507386\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.452507386\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.39 - 0.238i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-2.37 - 1.16i)T \) |
good | 11 | \( 1 - 4.86iT - 11T^{2} \) |
| 13 | \( 1 - 3.63iT - 13T^{2} \) |
| 17 | \( 1 + 4.47iT - 17T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 + 1.68iT - 23T^{2} \) |
| 29 | \( 1 + 8.31T + 29T^{2} \) |
| 31 | \( 1 - 5.47T + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 + 11.5iT - 41T^{2} \) |
| 43 | \( 1 - 7.86iT - 43T^{2} \) |
| 47 | \( 1 + 4.75T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 2.97T + 59T^{2} \) |
| 61 | \( 1 + 1.18iT - 61T^{2} \) |
| 67 | \( 1 + 13.1iT - 67T^{2} \) |
| 71 | \( 1 + 14.8iT - 71T^{2} \) |
| 73 | \( 1 - 1.40iT - 73T^{2} \) |
| 79 | \( 1 + 1.01iT - 79T^{2} \) |
| 83 | \( 1 + 8.22T + 83T^{2} \) |
| 89 | \( 1 - 9.91iT - 89T^{2} \) |
| 97 | \( 1 + 13.0iT - 97T^{2} \) |
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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
−9.725912625510262175473531037476, −8.968692413573786914012473849311, −7.916044835011438814290523913286, −7.26299089735021938916345619294, −6.41726179798626405739478603133, −5.32769988523032345377715024050, −4.65574290307953717086453150566, −4.09210652733536651418420209508, −2.46555476664511919811662163005, −1.75758187506658964704086278652,
1.16736127162490677675512145037, 2.52231917601838909286468853535, 3.54499647119421465512507215740, 4.26858368963974929099608994610, 5.51292998045502959515314335497, 5.95154237114767073983831062597, 6.98453471610596567218264563152, 7.956666268352612871699332922056, 8.451306081508967812091243537096, 9.993469800615353059257978788075