| 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 | \( 1 + (-3 + 3i)T - 97iT^{2} \) |
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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
−10.29114244288198554202836159454, −9.751561531843854565132887073834, −9.056792251200399634530463683178, −7.54254681243446590685683358450, −6.77680705173972987379947169767, −5.69385524670392918644676982756, −5.08995377658841635889391485991, −3.98157943307833483926616460479, −3.22293548106315928613307069977, −0.798051517344053497583917184647,
0.875958458013331180292750983529, 2.26682324829930236713447204304, 3.66404240530070806974758690128, 5.28655135622173376336904391238, 5.94787183488346952596562801812, 6.44681744869604256638195753490, 7.56452075924571639174983515610, 8.206740596615492220000420459678, 9.465245993447043182873358665702, 10.29261216253602200569395796978
