L(s) = 1 | + (−2 − 2i)3-s + (−2 + 2i)7-s + 5i·9-s + (1 − i)13-s + (5 + 5i)17-s + 4·19-s + 8·21-s + (−2 − 2i)23-s + (4 − 4i)27-s − 4i·29-s − 4i·31-s + (−1 − i)37-s − 4·39-s + (6 + 6i)43-s + (2 − 2i)47-s + ⋯ |
L(s) = 1 | + (−1.15 − 1.15i)3-s + (−0.755 + 0.755i)7-s + 1.66i·9-s + (0.277 − 0.277i)13-s + (1.21 + 1.21i)17-s + 0.917·19-s + 1.74·21-s + (−0.417 − 0.417i)23-s + (0.769 − 0.769i)27-s − 0.742i·29-s − 0.718i·31-s + (−0.164 − 0.164i)37-s − 0.640·39-s + (0.914 + 0.914i)43-s + (0.291 − 0.291i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.877269 - 0.249213i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.877269 - 0.249213i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (2 + 2i)T + 3iT^{2} \) |
| 7 | \( 1 + (2 - 2i)T - 7iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-1 + i)T - 13iT^{2} \) |
| 17 | \( 1 + (-5 - 5i)T + 17iT^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (2 + 2i)T + 23iT^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + (1 + i)T + 37iT^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + (-6 - 6i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2 + 2i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7 + 7i)T - 53iT^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + (-10 + 10i)T - 67iT^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + (-3 + 3i)T - 73iT^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + (-2 - 2i)T + 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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
−10.29261216253602200569395796978, −9.465245993447043182873358665702, −8.206740596615492220000420459678, −7.56452075924571639174983515610, −6.44681744869604256638195753490, −5.94787183488346952596562801812, −5.28655135622173376336904391238, −3.66404240530070806974758690128, −2.26682324829930236713447204304, −0.875958458013331180292750983529,
0.798051517344053497583917184647, 3.22293548106315928613307069977, 3.98157943307833483926616460479, 5.08995377658841635889391485991, 5.69385524670392918644676982756, 6.77680705173972987379947169767, 7.54254681243446590685683358450, 9.056792251200399634530463683178, 9.751561531843854565132887073834, 10.29114244288198554202836159454