L(s) = 1 | + (1.41 − 1.41i)5-s − 7-s + (−2.82 − 2.82i)11-s + (4 − 4i)13-s + 5.65i·17-s + (−4 − 4i)19-s − 1.41i·23-s + 0.999i·25-s + (−5.65 − 5.65i)29-s + (−1.41 + 1.41i)35-s + (−5 − 5i)37-s + 8.48·41-s + (−7 + 7i)43-s + 49-s + (−7.07 + 7.07i)53-s + ⋯ |
L(s) = 1 | + (0.632 − 0.632i)5-s − 0.377·7-s + (−0.852 − 0.852i)11-s + (1.10 − 1.10i)13-s + 1.37i·17-s + (−0.917 − 0.917i)19-s − 0.294i·23-s + 0.199i·25-s + (−1.05 − 1.05i)29-s + (−0.239 + 0.239i)35-s + (−0.821 − 0.821i)37-s + 1.32·41-s + (−1.06 + 1.06i)43-s + 0.142·49-s + (−0.971 + 0.971i)53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8642862004\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8642862004\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + (-1.41 + 1.41i)T - 5iT^{2} \) |
| 11 | \( 1 + (2.82 + 2.82i)T + 11iT^{2} \) |
| 13 | \( 1 + (-4 + 4i)T - 13iT^{2} \) |
| 17 | \( 1 - 5.65iT - 17T^{2} \) |
| 19 | \( 1 + (4 + 4i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.41iT - 23T^{2} \) |
| 29 | \( 1 + (5.65 + 5.65i)T + 29iT^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (5 + 5i)T + 37iT^{2} \) |
| 41 | \( 1 - 8.48T + 41T^{2} \) |
| 43 | \( 1 + (7 - 7i)T - 43iT^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (7.07 - 7.07i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.65 - 5.65i)T + 59iT^{2} \) |
| 61 | \( 1 + (-4 + 4i)T - 61iT^{2} \) |
| 67 | \( 1 + (7 + 7i)T + 67iT^{2} \) |
| 71 | \( 1 - 7.07iT - 71T^{2} \) |
| 73 | \( 1 + 6iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + (-2.82 + 2.82i)T - 83iT^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.296338831577596944535804624470, −7.52652767179283887558006735321, −6.30301137118999972768284539117, −5.90941006466283723664453798427, −5.33028258093053417328860863069, −4.25843238954037159629405639631, −3.42291226016467968963239975250, −2.51834235423621424738422275888, −1.40808762023501475524035457687, −0.23237381149250893244111173347,
1.62888437451601816455697276571, 2.35494525930556079659087333845, 3.32701044593420191966898336174, 4.18352806242583437148571878432, 5.14828387858008289339494625226, 5.86305698738240146533915991268, 6.78602157802354408076369079953, 7.00913457909608887097481859499, 8.064612073861997764149281254851, 8.842530863360369858307504697043