L(s) = 1 | − i·2-s − 4-s + i·8-s + (−1 − i)11-s + 16-s − i·17-s − 2i·19-s + (−1 + i)22-s − i·25-s − i·32-s − 34-s − 2·38-s + (1 + i)41-s + (1 + i)44-s + i·49-s − 50-s + ⋯ |
L(s) = 1 | − i·2-s − 4-s + i·8-s + (−1 − i)11-s + 16-s − i·17-s − 2i·19-s + (−1 + i)22-s − i·25-s − i·32-s − 34-s − 2·38-s + (1 + i)41-s + (1 + i)44-s + i·49-s − 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7850932496\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7850932496\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 17 | \( 1 + iT \) |
good | 5 | \( 1 + iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (1 + i)T + iT^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 + 2iT - T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + iT^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-1 - i)T + iT^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + (1 - i)T - iT^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-1 + i)T - iT^{2} \) |
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
−9.649872043138893290139805678803, −8.950611728422764908099295445106, −8.212032346760379786178248673044, −7.30204172044049002920977484137, −6.06687127055924960334419447481, −5.08855432975777342336348347288, −4.41299233458646797171435577354, −3.01715222668309592888988957221, −2.53746140348679048091065090571, −0.70171238823026773156168773110,
1.80829032900124478055635881818, 3.49234921274683852657239782212, 4.36608301691084126756099485302, 5.41116395471109018874759007869, 5.98024557853827041292543309453, 7.08370308148093357864198090091, 7.75649622329452560886027512087, 8.357070239103910472158559628683, 9.356461625409472482903509700555, 10.14308166211246294109234134243