L(s) = 1 | + (0.803 − 0.595i)2-s + (0.290 − 0.956i)4-s + (−0.427 − 0.903i)7-s + (−0.336 − 0.941i)8-s + (0.998 + 0.0490i)9-s + (1.14 + 1.63i)11-s + (−0.881 − 0.471i)14-s + (−0.831 − 0.555i)16-s + (0.831 − 0.555i)18-s + (1.89 + 0.625i)22-s + (1.07 + 0.800i)23-s + (0.857 + 0.514i)25-s + (−0.989 + 0.146i)28-s + (0.215 − 0.961i)29-s + (−0.998 + 0.0490i)32-s + ⋯ |
L(s) = 1 | + (0.803 − 0.595i)2-s + (0.290 − 0.956i)4-s + (−0.427 − 0.903i)7-s + (−0.336 − 0.941i)8-s + (0.998 + 0.0490i)9-s + (1.14 + 1.63i)11-s + (−0.881 − 0.471i)14-s + (−0.831 − 0.555i)16-s + (0.831 − 0.555i)18-s + (1.89 + 0.625i)22-s + (1.07 + 0.800i)23-s + (0.857 + 0.514i)25-s + (−0.989 + 0.146i)28-s + (0.215 − 0.961i)29-s + (−0.998 + 0.0490i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.278 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.227983895\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.227983895\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.803 + 0.595i)T \) |
| 7 | \( 1 + (0.427 + 0.903i)T \) |
good | 3 | \( 1 + (-0.998 - 0.0490i)T^{2} \) |
| 5 | \( 1 + (-0.857 - 0.514i)T^{2} \) |
| 11 | \( 1 + (-1.14 - 1.63i)T + (-0.336 + 0.941i)T^{2} \) |
| 13 | \( 1 + (0.970 + 0.242i)T^{2} \) |
| 17 | \( 1 + (0.831 - 0.555i)T^{2} \) |
| 19 | \( 1 + (-0.989 + 0.146i)T^{2} \) |
| 23 | \( 1 + (-1.07 - 0.800i)T + (0.290 + 0.956i)T^{2} \) |
| 29 | \( 1 + (-0.215 + 0.961i)T + (-0.903 - 0.427i)T^{2} \) |
| 31 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 37 | \( 1 + (1.02 + 0.517i)T + (0.595 + 0.803i)T^{2} \) |
| 41 | \( 1 + (-0.471 + 0.881i)T^{2} \) |
| 43 | \( 1 + (1.22 + 1.16i)T + (0.0490 + 0.998i)T^{2} \) |
| 47 | \( 1 + (-0.195 + 0.980i)T^{2} \) |
| 53 | \( 1 + (0.408 + 1.82i)T + (-0.903 + 0.427i)T^{2} \) |
| 59 | \( 1 + (-0.970 + 0.242i)T^{2} \) |
| 61 | \( 1 + (-0.671 - 0.740i)T^{2} \) |
| 67 | \( 1 + (1.28 - 0.496i)T + (0.740 - 0.671i)T^{2} \) |
| 71 | \( 1 + (1.33 - 1.47i)T + (-0.0980 - 0.995i)T^{2} \) |
| 73 | \( 1 + (0.634 + 0.773i)T^{2} \) |
| 79 | \( 1 + (-0.0476 - 0.483i)T + (-0.980 + 0.195i)T^{2} \) |
| 83 | \( 1 + (-0.595 + 0.803i)T^{2} \) |
| 89 | \( 1 + (-0.290 + 0.956i)T^{2} \) |
| 97 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
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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
−8.857458843983142626012926055082, −7.33917772909827459504778758014, −7.03552095541102312039029661720, −6.53753190257282131746720299186, −5.28621349580090201066369047184, −4.56415561045886630590913795275, −3.98564542534075536011555998319, −3.28348282419300503762232626284, −1.92909758223065610027628329124, −1.23311566180158444325185888537,
1.44073596919331247313632739280, 2.97104328664913127094931506345, 3.31531119684530620926513498704, 4.44201223419739731111277223474, 5.07462672003298444033386507857, 6.11949981125297456672220272790, 6.45745986624168321786154163851, 7.11880900342986539301653634654, 8.173548945419554493879052164138, 8.901087374153088399082248383660