L(s) = 1 | + (1.93 + 1.11i)3-s + (1.11 + 1.93i)5-s + (−1.73 + 2i)7-s + (1 + 1.73i)9-s + (1.93 − 3.35i)11-s + 5.00i·15-s + (4.5 + 2.59i)17-s + (−5.80 + 3.35i)19-s + (−5.59 + 1.93i)21-s + (−2.59 + 1.5i)23-s − 2.23i·27-s − 7.74i·29-s + (−0.866 + 1.5i)31-s + (7.50 − 4.33i)33-s + (−5.80 − 1.11i)35-s + ⋯ |
L(s) = 1 | + (1.11 + 0.645i)3-s + (0.499 + 0.866i)5-s + (−0.654 + 0.755i)7-s + (0.333 + 0.577i)9-s + (0.583 − 1.01i)11-s + 1.29i·15-s + (1.09 + 0.630i)17-s + (−1.33 + 0.769i)19-s + (−1.21 + 0.422i)21-s + (−0.541 + 0.312i)23-s − 0.430i·27-s − 1.43i·29-s + (−0.155 + 0.269i)31-s + (1.30 − 0.753i)33-s + (−0.981 − 0.188i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.64408 + 1.18072i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.64408 + 1.18072i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
good | 3 | \( 1 + (-1.93 - 1.11i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.11 - 1.93i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.93 + 3.35i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-4.5 - 2.59i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.80 - 3.35i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 - 1.5i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 7.74iT - 29T^{2} \) |
| 31 | \( 1 + (0.866 - 1.5i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-10.0 + 5.80i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + (2.59 + 4.5i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.35 + 1.93i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.93 - 1.11i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.35 - 5.80i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.80 + 10.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-11.2 + 6.5i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 4.47iT - 83T^{2} \) |
| 89 | \( 1 + (4.5 - 2.59i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.46iT - 97T^{2} \) |
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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
−11.09628108200100819981072472884, −10.01543102389889640475487091498, −9.650701195430020639131156216547, −8.586899285714230444455840732871, −7.974147928900978107498816499581, −6.28255872261436343702920046575, −5.97161482540490519470992003754, −4.02455818587310813697000865993, −3.23081911973860292364268492628, −2.28154425166454555859248951814,
1.29159057992756876896181853998, 2.56323623286910209665701306601, 3.89810099323124256262399775153, 5.04324770388531619698357769560, 6.53204185511755639432788566654, 7.28615239482416956236082109458, 8.214464167539633558052896999881, 9.194122689962226699195035675820, 9.651922646554306939372907959860, 10.76200981458468319376060226356