| L(s) = 1 | − 9·7-s − 18.3i·11-s − 13-s − 26.8i·17-s − 25·19-s − 4.24i·23-s − 26.8i·29-s + 39·31-s − 32·37-s − 5.65i·41-s − 23·43-s − 32.5i·47-s + 32·49-s + 96.1i·53-s − 9.89i·59-s + ⋯ |
| L(s) = 1 | − 1.28·7-s − 1.67i·11-s − 0.0769·13-s − 1.58i·17-s − 1.31·19-s − 0.184i·23-s − 0.926i·29-s + 1.25·31-s − 0.864·37-s − 0.137i·41-s − 0.534·43-s − 0.692i·47-s + 0.653·49-s + 1.81i·53-s − 0.167i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1977786871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1977786871\) |
| \(L(2)\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 9T + 49T^{2} \) |
| 11 | \( 1 + 18.3iT - 121T^{2} \) |
| 13 | \( 1 + T + 169T^{2} \) |
| 17 | \( 1 + 26.8iT - 289T^{2} \) |
| 19 | \( 1 + 25T + 361T^{2} \) |
| 23 | \( 1 + 4.24iT - 529T^{2} \) |
| 29 | \( 1 + 26.8iT - 841T^{2} \) |
| 31 | \( 1 - 39T + 961T^{2} \) |
| 37 | \( 1 + 32T + 1.36e3T^{2} \) |
| 41 | \( 1 + 5.65iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23T + 1.84e3T^{2} \) |
| 47 | \( 1 + 32.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 96.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 9.89iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 73T + 3.72e3T^{2} \) |
| 67 | \( 1 - 63T + 4.48e3T^{2} \) |
| 71 | \( 1 + 62.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 136T + 5.32e3T^{2} \) |
| 79 | \( 1 - 24T + 6.24e3T^{2} \) |
| 83 | \( 1 + 46.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 7T + 9.40e3T^{2} \) |
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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
−7.997636132580568519118650313442, −6.98561126807676597054637360868, −6.38316475790437159515744164872, −5.83379030060162939759408543609, −4.86550632116231161790535574079, −3.86797625280582361817074448983, −3.08086048173263521894365911046, −2.44890198383562394385072468004, −0.77087749458354807650065673550, −0.05523172074852043384134555270,
1.57228583041082317301656131480, 2.41276336916997655515474657751, 3.49382943819530255912410999753, 4.19810831041191388595881267396, 5.01557329520137064972930054782, 6.09129067577034291958928698301, 6.65997258995402939772140314164, 7.17135805382902305190067607425, 8.252796158568762970501735270187, 8.776138061745164346003530987262
