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
−10.19183224310111715113604163061, −8.685725660314907289695897570067, −7.78290228155450125071304856516, −7.49348495787052995878366964767, −5.95355639896839294998784020101, −5.44947177375869956051496625855, −4.60298063381152501560751660433, −3.95729406706461226334667497226, −2.60845091583456539254983635707, −1.03665851419712475305052226993,
0.62456331014394869521802831809, 2.02011922464313568904388488612, 2.85247097279765507552717190228, 4.58748865239282000002022218887, 5.24412361937650528365172893425, 5.89229609099618190860622162201, 6.86460401329444364478919878263, 7.64923158087932122910824507750, 8.175626513430294727671704552172, 9.367439907011658074605770595255