L(s) = 1 | − 2.24i·3-s + 3.30i·5-s − 7-s − 2.05·9-s − 0.941i·11-s − 5.19i·13-s + 7.43·15-s − 6.49·17-s + 1.75i·19-s + 2.24i·21-s + 5.55·23-s − 5.94·25-s − 2.11i·27-s + 2.49i·29-s + 6.61·31-s + ⋯ |
L(s) = 1 | − 1.29i·3-s + 1.47i·5-s − 0.377·7-s − 0.686·9-s − 0.283i·11-s − 1.43i·13-s + 1.92·15-s − 1.57·17-s + 0.401i·19-s + 0.490i·21-s + 1.15·23-s − 1.18·25-s − 0.407i·27-s + 0.463i·29-s + 1.18·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.022437240\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.022437240\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + 2.24iT - 3T^{2} \) |
| 5 | \( 1 - 3.30iT - 5T^{2} \) |
| 11 | \( 1 + 0.941iT - 11T^{2} \) |
| 13 | \( 1 + 5.19iT - 13T^{2} \) |
| 17 | \( 1 + 6.49T + 17T^{2} \) |
| 19 | \( 1 - 1.75iT - 19T^{2} \) |
| 23 | \( 1 - 5.55T + 23T^{2} \) |
| 29 | \( 1 - 2.49iT - 29T^{2} \) |
| 31 | \( 1 - 6.61T + 31T^{2} \) |
| 37 | \( 1 + 4.61iT - 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 43 | \( 1 + 12.0iT - 43T^{2} \) |
| 47 | \( 1 + 4.49T + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 + 12.6iT - 59T^{2} \) |
| 61 | \( 1 + 11.3iT - 61T^{2} \) |
| 67 | \( 1 + 0.443iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 0.117T + 73T^{2} \) |
| 79 | \( 1 + 7.11T + 79T^{2} \) |
| 83 | \( 1 + 12.8iT - 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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
−8.737677253534684584415935789206, −8.031294259422368864381514686766, −7.17414684623057998593584310752, −6.71553916664298410304340274875, −6.18129228605614001216435163172, −5.09544184595677539738701435384, −3.54106236384921421041604130961, −2.84883514960626340750547620852, −1.96326954659058926950274484669, −0.38118283706441094932047864593,
1.40247719937009744468492574690, 2.83416325579896098053797702183, 4.26014003341430642980449274631, 4.47941177318076962296230761753, 5.11064404078844423582637019979, 6.33368593614238183667886395469, 7.09944565831461447770131909899, 8.613212870847648758136077079099, 8.764658203050112335251966288934, 9.574997821027663582470648515605