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
−9.993469800615353059257978788075, −8.451306081508967812091243537096, −7.956666268352612871699332922056, −6.98453471610596567218264563152, −5.95154237114767073983831062597, −5.51292998045502959515314335497, −4.26858368963974929099608994610, −3.54499647119421465512507215740, −2.52231917601838909286468853535, −1.16736127162490677675512145037,
1.75758187506658964704086278652, 2.46555476664511919811662163005, 4.09210652733536651418420209508, 4.65574290307953717086453150566, 5.32769988523032345377715024050, 6.41726179798626405739478603133, 7.26299089735021938916345619294, 7.916044835011438814290523913286, 8.968692413573786914012473849311, 9.725912625510262175473531037476