L(s) = 1 | + 4i·7-s + 16.9i·11-s − 8i·13-s − 12.7·17-s − 16·19-s − 16.9·23-s + 4.24i·29-s − 44·31-s − 34i·37-s − 46.6i·41-s − 40i·43-s + 84.8·47-s + 33·49-s − 38.1·53-s − 33.9i·59-s + ⋯ |
L(s) = 1 | + 0.571i·7-s + 1.54i·11-s − 0.615i·13-s − 0.748·17-s − 0.842·19-s − 0.737·23-s + 0.146i·29-s − 1.41·31-s − 0.918i·37-s − 1.13i·41-s − 0.930i·43-s + 1.80·47-s + 0.673·49-s − 0.720·53-s − 0.575i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.107117711\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.107117711\) |
\(L(2)\) |
|
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 | \( 1 - 4iT - 49T^{2} \) |
| 11 | \( 1 - 16.9iT - 121T^{2} \) |
| 13 | \( 1 + 8iT - 169T^{2} \) |
| 17 | \( 1 + 12.7T + 289T^{2} \) |
| 19 | \( 1 + 16T + 361T^{2} \) |
| 23 | \( 1 + 16.9T + 529T^{2} \) |
| 29 | \( 1 - 4.24iT - 841T^{2} \) |
| 31 | \( 1 + 44T + 961T^{2} \) |
| 37 | \( 1 + 34iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 46.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 40iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 84.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 38.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 50T + 3.72e3T^{2} \) |
| 67 | \( 1 + 8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 16iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 76T + 6.24e3T^{2} \) |
| 83 | \( 1 - 118.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 12.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 176iT - 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.294260703176643198806978171045, −7.39263627608952614784292180794, −6.93454743640997610207329824872, −5.90895261288427929672620154760, −5.30072665077767955990513057722, −4.37292885685795860683821094097, −3.70208875443575326618600155733, −2.30777988853573116714202281444, −1.98079620785250773521164167292, −0.28764384775761840503409300458,
0.815549620312793193444007062190, 1.97235100374910980055079660372, 3.02495812600685257221240920940, 3.94155193419132361395351100578, 4.51748468176779986932623218938, 5.67073664775078830912228929074, 6.25192813680709348381649008062, 6.97601354579089690894313090741, 7.82720586428621170746411707581, 8.565095037515012308826471427369