L(s) = 1 | + (−1.41 − 1.41i)3-s + (2.44 + 2.44i)7-s + 1.00i·9-s − 3.46·11-s + (−2.82 + 2.82i)13-s + (4.89 − 4.89i)17-s − 3.46i·19-s − 6.92i·21-s + (−2.44 + 2.44i)23-s + (−2.82 + 2.82i)27-s + 6.92·29-s − 8i·31-s + (4.89 + 4.89i)33-s + (5.65 + 5.65i)37-s + 8.00·39-s + ⋯ |
L(s) = 1 | + (−0.816 − 0.816i)3-s + (0.925 + 0.925i)7-s + 0.333i·9-s − 1.04·11-s + (−0.784 + 0.784i)13-s + (1.18 − 1.18i)17-s − 0.794i·19-s − 1.51i·21-s + (−0.510 + 0.510i)23-s + (−0.544 + 0.544i)27-s + 1.28·29-s − 1.43i·31-s + (0.852 + 0.852i)33-s + (0.929 + 0.929i)37-s + 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.174445769\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.174445769\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.41 + 1.41i)T + 3iT^{2} \) |
| 7 | \( 1 + (-2.44 - 2.44i)T + 7iT^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + (2.82 - 2.82i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.89 + 4.89i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + (2.44 - 2.44i)T - 23iT^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 8iT - 31T^{2} \) |
| 37 | \( 1 + (-5.65 - 5.65i)T + 37iT^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + (-1.41 - 1.41i)T + 43iT^{2} \) |
| 47 | \( 1 + (-2.44 - 2.44i)T + 47iT^{2} \) |
| 53 | \( 1 + (-8.48 + 8.48i)T - 53iT^{2} \) |
| 59 | \( 1 + 10.3iT - 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 + (1.41 - 1.41i)T - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (4.89 + 4.89i)T + 73iT^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + (-4.24 - 4.24i)T + 83iT^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + (4.89 - 4.89i)T - 97iT^{2} \) |
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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
−9.367439907011658074605770595255, −8.175626513430294727671704552172, −7.64923158087932122910824507750, −6.86460401329444364478919878263, −5.89229609099618190860622162201, −5.24412361937650528365172893425, −4.58748865239282000002022218887, −2.85247097279765507552717190228, −2.02011922464313568904388488612, −0.62456331014394869521802831809,
1.03665851419712475305052226993, 2.60845091583456539254983635707, 3.95729406706461226334667497226, 4.60298063381152501560751660433, 5.44947177375869956051496625855, 5.95355639896839294998784020101, 7.49348495787052995878366964767, 7.78290228155450125071304856516, 8.685725660314907289695897570067, 10.19183224310111715113604163061