| 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
−8.842530863360369858307504697043, −8.064612073861997764149281254851, −7.00913457909608887097481859499, −6.78602157802354408076369079953, −5.86305698738240146533915991268, −5.14828387858008289339494625226, −4.18352806242583437148571878432, −3.32701044593420191966898336174, −2.35494525930556079659087333845, −1.62888437451601816455697276571,
0.23237381149250893244111173347, 1.40808762023501475524035457687, 2.51834235423621424738422275888, 3.42291226016467968963239975250, 4.25843238954037159629405639631, 5.33028258093053417328860863069, 5.90941006466283723664453798427, 6.30301137118999972768284539117, 7.52652767179283887558006735321, 8.296338831577596944535804624470
