| L(s) = 1 | + 4.29·5-s + 11.2i·7-s − 8.53i·13-s + 18.8i·17-s − 19.4i·19-s − 34.9·23-s − 6.54·25-s − 9.50i·29-s + 24.6·31-s + 48.4i·35-s − 27.0·37-s − 57.3i·41-s − 41.2i·43-s − 53.4·47-s − 77.9·49-s + ⋯ |
| L(s) = 1 | + 0.859·5-s + 1.60i·7-s − 0.656i·13-s + 1.10i·17-s − 1.02i·19-s − 1.51·23-s − 0.261·25-s − 0.327i·29-s + 0.795·31-s + 1.38i·35-s − 0.729·37-s − 1.39i·41-s − 0.960i·43-s − 1.13·47-s − 1.59·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5347182879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5347182879\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 4.29T + 25T^{2} \) |
| 7 | \( 1 - 11.2iT - 49T^{2} \) |
| 13 | \( 1 + 8.53iT - 169T^{2} \) |
| 17 | \( 1 - 18.8iT - 289T^{2} \) |
| 19 | \( 1 + 19.4iT - 361T^{2} \) |
| 23 | \( 1 + 34.9T + 529T^{2} \) |
| 29 | \( 1 + 9.50iT - 841T^{2} \) |
| 31 | \( 1 - 24.6T + 961T^{2} \) |
| 37 | \( 1 + 27.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 57.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 41.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 53.4T + 2.20e3T^{2} \) |
| 53 | \( 1 + 25.0T + 2.80e3T^{2} \) |
| 59 | \( 1 - 101.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 114. iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 73.4T + 4.48e3T^{2} \) |
| 71 | \( 1 + 105.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 41.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 76.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 129. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 154.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 106.T + 9.40e3T^{2} \) |
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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
−8.192421002138531343827415758496, −7.19547285339268762869751226903, −6.13822850832243276416984348113, −5.86518775246535896888979812208, −5.23717047198930622274377684412, −4.22062581862698912435206832121, −3.12240543079087497860629780131, −2.28244150464099346779621522883, −1.73038219387012597949989344031, −0.099836386686913779054385416130,
1.17378724881367584830920623909, 1.91053729800837299093909466692, 3.08699497882799566745225789578, 4.00141826374762254325990953547, 4.60764178194465475636772517152, 5.53178943389692319919412912715, 6.42837756183423357773500387525, 6.84803087360059000843481662371, 7.78773828839549266946116367741, 8.230622428735184849169163627632
