L(s) = 1 | + (1.40 + 0.192i)2-s + (1.92 + 0.540i)4-s + 0.0802·7-s + (2.59 + 1.12i)8-s − 2.41i·11-s − 5.26i·13-s + (0.112 + 0.0154i)14-s + (3.41 + 2.08i)16-s + 0.255·17-s − 6.95i·19-s + (0.465 − 3.38i)22-s + 1.64·23-s + (1.01 − 7.38i)26-s + (0.154 + 0.0433i)28-s + 4.51i·29-s + ⋯ |
L(s) = 1 | + (0.990 + 0.136i)2-s + (0.962 + 0.270i)4-s + 0.0303·7-s + (0.917 + 0.398i)8-s − 0.728i·11-s − 1.46i·13-s + (0.0300 + 0.00413i)14-s + (0.854 + 0.520i)16-s + 0.0620·17-s − 1.59i·19-s + (0.0993 − 0.721i)22-s + 0.343·23-s + (0.199 − 1.44i)26-s + (0.0292 + 0.00819i)28-s + 0.838i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.464738743\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.464738743\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 - 0.192i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.0802T + 7T^{2} \) |
| 11 | \( 1 + 2.41iT - 11T^{2} \) |
| 13 | \( 1 + 5.26iT - 13T^{2} \) |
| 17 | \( 1 - 0.255T + 17T^{2} \) |
| 19 | \( 1 + 6.95iT - 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 - 4.51iT - 29T^{2} \) |
| 31 | \( 1 - 8.29T + 31T^{2} \) |
| 37 | \( 1 + 2.67iT - 37T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 43 | \( 1 - 4.08iT - 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 - 11.5iT - 53T^{2} \) |
| 59 | \( 1 - 12.6iT - 59T^{2} \) |
| 61 | \( 1 + 11.9iT - 61T^{2} \) |
| 67 | \( 1 - 7.27iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 5.50T + 79T^{2} \) |
| 83 | \( 1 + 9.20iT - 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 - 8.50T + 97T^{2} \) |
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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
−9.129807525979361088159332257041, −8.231255297193905007836330105688, −7.56414135202224173097404345508, −6.67045496837155846170312380030, −5.90455143057892867622927379275, −5.13892577899517432454484286854, −4.38536839837063685618974929103, −3.13617745266672618062060462286, −2.71928462659492097950351016118, −0.999364689653223330140282786883,
1.51745866696070930834312763771, 2.37803003790003598869002719938, 3.62419051752621337495728488838, 4.35541360107539058903010222580, 5.08456330194721807641326234404, 6.18973290649729129670855026227, 6.66975675740334960900995220728, 7.60620483066491902726100936647, 8.371999876454533937125262621884, 9.690775350124190188225719956950