| L(s) = 1 | + (−1.90 + 4.62i)5-s + 0.764i·7-s − 8.06i·11-s − 2.16i·13-s − 24.9·17-s + 20.7·19-s + 16.8·23-s + (−17.7 − 17.6i)25-s − 11.7i·29-s + 0.669·31-s + (−3.53 − 1.45i)35-s + 48.2i·37-s − 62.5i·41-s − 71.3i·43-s + 20.2·47-s + ⋯ |
| L(s) = 1 | + (−0.381 + 0.924i)5-s + 0.109i·7-s − 0.733i·11-s − 0.166i·13-s − 1.46·17-s + 1.09·19-s + 0.732·23-s + (−0.708 − 0.705i)25-s − 0.404i·29-s + 0.0215·31-s + (−0.100 − 0.0417i)35-s + 1.30i·37-s − 1.52i·41-s − 1.65i·43-s + 0.431·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.533325499\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533325499\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.90 - 4.62i)T \) |
| good | 7 | \( 1 - 0.764iT - 49T^{2} \) |
| 11 | \( 1 + 8.06iT - 121T^{2} \) |
| 13 | \( 1 + 2.16iT - 169T^{2} \) |
| 17 | \( 1 + 24.9T + 289T^{2} \) |
| 19 | \( 1 - 20.7T + 361T^{2} \) |
| 23 | \( 1 - 16.8T + 529T^{2} \) |
| 29 | \( 1 + 11.7iT - 841T^{2} \) |
| 31 | \( 1 - 0.669T + 961T^{2} \) |
| 37 | \( 1 - 48.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 62.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 71.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 20.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 82.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 5.39iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 83.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 10.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 87.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 15.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 33.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 55.7T + 6.88e3T^{2} \) |
| 89 | \( 1 + 22.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 68.9iT - 9.40e3T^{2} \) |
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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
−9.046197996554900024164846145894, −8.423788221821474476060428238966, −7.42733731803334626772613338331, −6.84428201035480362194176689137, −6.00138764482124780778651509967, −5.06836244435034394452256969501, −3.94023663943922022696603101034, −3.13269195919750757470192471917, −2.20610340794361409810169784686, −0.54380002321230363584664257666,
0.874502141253319876888809222224, 2.03464772638181315690901327153, 3.32855271900513297555256361967, 4.48098427950758860365174352157, 4.86530892304555534002856833970, 5.97770099669830092459449268079, 7.00156731787976703799409371713, 7.64535841365609246717856132670, 8.538908402213741814554264943607, 9.277979564826777646758751063623
