L(s) = 1 | − 3-s + i·5-s − i·7-s + 9-s + 3i·11-s + (3 + 2i)13-s − i·15-s + 7·17-s + 3i·19-s + i·21-s − 23-s + 4·25-s − 27-s − 29-s − 8i·31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447i·5-s − 0.377i·7-s + 0.333·9-s + 0.904i·11-s + (0.832 + 0.554i)13-s − 0.258i·15-s + 1.69·17-s + 0.688i·19-s + 0.218i·21-s − 0.208·23-s + 0.800·25-s − 0.192·27-s − 0.185·29-s − 1.43i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.690887109\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.690887109\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (-3 - 2i)T \) |
good | 5 | \( 1 - iT - 5T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 19 | \( 1 - 3iT - 19T^{2} \) |
| 23 | \( 1 + T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 - iT - 37T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 - 13iT - 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 2iT - 83T^{2} \) |
| 89 | \( 1 + 12iT - 89T^{2} \) |
| 97 | \( 1 + 6iT - 97T^{2} \) |
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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
−8.349767129780511638535051620321, −7.54664289122515777609877662114, −7.13675936219224482208945724592, −6.15677583157293307219287170823, −5.74480894316983856240713556968, −4.69560488741758750890461214685, −3.98620155066313285100279508886, −3.19281409100618000644348155891, −1.94683708157462223591076479602, −0.995126112174070782269970225459,
0.66854931075850297804195872741, 1.46423731735087239600263554104, 2.99781194192239128282687873459, 3.52127693754453556156401590873, 4.75338849251651190930964375180, 5.33017364760809383556233696477, 5.98520933629242487396428346402, 6.59892478696932866614096836323, 7.67485510987562422449112944898, 8.209662568024948702209953936941