L(s) = 1 | − i·2-s − 3-s − 4-s − i·5-s + i·6-s + i·8-s + 9-s − 10-s − 2i·11-s + 12-s + 13-s + i·15-s + 16-s − i·18-s + i·20-s + ⋯ |
L(s) = 1 | − i·2-s − 3-s − 4-s − i·5-s + i·6-s + i·8-s + 9-s − 10-s − 2i·11-s + 12-s + 13-s + i·15-s + 16-s − i·18-s + i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6371074505\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6371074505\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + 2iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 2T + T^{2} \) |
| 47 | \( 1 + 2iT - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 2iT - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - 2T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.280415348874297448936626358500, −8.570554956129462779228584161184, −8.047314160610133101530080005359, −6.49147006486127271139307636863, −5.65791256348840299180803701653, −5.14637900411488151018799580618, −4.05881082455783942215491415621, −3.33109429810329716051787720410, −1.60489032569598657217205765218, −0.62866525652835745724448548805,
1.72960181570808695176212245361, 3.51548440099056782291667955892, 4.47929764363723737965942924075, 5.17028030406533126560967941397, 6.31658448585394989182106715003, 6.61618216905513338139999218469, 7.41310843047231011074407866626, 8.076897121303921053203726152889, 9.465881515785681660784069950311, 9.891767780930705834507708458391