L(s) = 1 | − 3.60i·3-s − 4·5-s + 5·7-s − 3.99·9-s − 10·11-s − 3.60i·13-s + 14.4i·15-s + 15·17-s + (−6 + 18.0i)19-s − 18.0i·21-s − 35·23-s − 9·25-s − 18.0i·27-s − 18.0i·29-s + 36.0i·31-s + ⋯ |
L(s) = 1 | − 1.20i·3-s − 0.800·5-s + 0.714·7-s − 0.444·9-s − 0.909·11-s − 0.277i·13-s + 0.961i·15-s + 0.882·17-s + (−0.315 + 0.948i)19-s − 0.858i·21-s − 1.52·23-s − 0.359·25-s − 0.667i·27-s − 0.621i·29-s + 1.16i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5839130913\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5839130913\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (6 - 18.0i)T \) |
good | 3 | \( 1 + 3.60iT - 9T^{2} \) |
| 5 | \( 1 + 4T + 25T^{2} \) |
| 7 | \( 1 - 5T + 49T^{2} \) |
| 11 | \( 1 + 10T + 121T^{2} \) |
| 13 | \( 1 + 3.60iT - 169T^{2} \) |
| 17 | \( 1 - 15T + 289T^{2} \) |
| 23 | \( 1 + 35T + 529T^{2} \) |
| 29 | \( 1 + 18.0iT - 841T^{2} \) |
| 31 | \( 1 - 36.0iT - 961T^{2} \) |
| 37 | \( 1 - 21.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 36.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 20T + 1.84e3T^{2} \) |
| 47 | \( 1 + 10T + 2.20e3T^{2} \) |
| 53 | \( 1 - 75.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 18.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 40T + 3.72e3T^{2} \) |
| 67 | \( 1 - 39.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 108. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 105T + 5.32e3T^{2} \) |
| 79 | \( 1 - 36.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 40T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 122. iT - 9.40e3T^{2} \) |
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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
−9.855705431160310149988817256651, −8.313943586375826253115996016706, −7.953874850726800222253321761387, −7.58252707996188553207210961000, −6.45442594173889976787522414484, −5.62871307630947176080034633845, −4.55960095006345851383012882988, −3.47817550771349350694678232705, −2.20983681933586124752378825093, −1.20854645194228079265580948691,
0.18093373546281496422424141430, 2.08797418286486006571866143206, 3.46198570509865113918864939610, 4.17253266087763365785599155327, 4.94434417578348280169650173334, 5.69627482715020209623018994046, 7.08250472108462689397100210186, 7.935097899905143739865265091968, 8.471060404125945718378618311284, 9.589567326250622356930278152546