| L(s) = 1 | − 2.59i·2-s − 2.71·4-s − 2.23·5-s − 0.880i·7-s − 3.31i·8-s + 5.79i·10-s + (−4.64 − 9.97i)11-s + 8.45i·13-s − 2.28·14-s − 19.4·16-s + 1.22i·17-s − 17.1i·19-s + 6.08·20-s + (−25.8 + 12.0i)22-s − 20.2·23-s + ⋯ |
| L(s) = 1 | − 1.29i·2-s − 0.679·4-s − 0.447·5-s − 0.125i·7-s − 0.414i·8-s + 0.579i·10-s + (−0.422 − 0.906i)11-s + 0.650i·13-s − 0.162·14-s − 1.21·16-s + 0.0718i·17-s − 0.903i·19-s + 0.304·20-s + (−1.17 + 0.547i)22-s − 0.880·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5889039388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5889039388\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
| 11 | \( 1 + (4.64 + 9.97i)T \) |
| good | 2 | \( 1 + 2.59iT - 4T^{2} \) |
| 7 | \( 1 + 0.880iT - 49T^{2} \) |
| 13 | \( 1 - 8.45iT - 169T^{2} \) |
| 17 | \( 1 - 1.22iT - 289T^{2} \) |
| 19 | \( 1 + 17.1iT - 361T^{2} \) |
| 23 | \( 1 + 20.2T + 529T^{2} \) |
| 29 | \( 1 - 35.3iT - 841T^{2} \) |
| 31 | \( 1 + 49.4T + 961T^{2} \) |
| 37 | \( 1 + 16.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 29.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 26.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 67.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 52.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 26.1T + 3.48e3T^{2} \) |
| 61 | \( 1 - 30.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 26.7T + 4.48e3T^{2} \) |
| 71 | \( 1 - 35.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 134. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 20.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 50.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 165.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 16.1T + 9.40e3T^{2} \) |
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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
−10.47436380321073816821249597073, −9.380951835632529550227101945035, −8.645491821821129074356590076166, −7.44285630616948661962450034058, −6.46279414208076615522148082622, −5.05622573738942578001853235326, −3.88453930490904530267967458047, −3.05873650184240482649744427192, −1.75036003389716651499582148387, −0.22323468683412905177045034947,
2.18248711323201794836469870965, 3.84267485695296079052702272068, 5.04264014178584163910878851306, 5.84058620999837948713335941822, 6.86069869631844189462406464765, 7.77848412516711779217157633581, 8.160599616755682592245388719964, 9.383950769456894444617241394917, 10.32437327798381587914638094495, 11.35986108451992197687197071385
