| L(s) = 1 | + (−1.09 − 0.894i)2-s + (−1.71 − 0.212i)3-s + (0.399 + 1.95i)4-s + (2.09 + 2.39i)5-s + (1.69 + 1.77i)6-s + (−0.401 − 0.0529i)7-s + (1.31 − 2.50i)8-s + (2.90 + 0.729i)9-s + (−0.158 − 4.49i)10-s + (−0.0136 − 0.0275i)11-s + (−0.271 − 3.45i)12-s + (0.691 + 2.03i)13-s + (0.392 + 0.417i)14-s + (−3.10 − 4.56i)15-s + (−3.68 + 1.56i)16-s + (−1.86 − 1.86i)17-s + ⋯ |
| L(s) = 1 | + (−0.774 − 0.632i)2-s + (−0.992 − 0.122i)3-s + (0.199 + 0.979i)4-s + (0.938 + 1.07i)5-s + (0.691 + 0.722i)6-s + (−0.151 − 0.0199i)7-s + (0.464 − 0.885i)8-s + (0.969 + 0.243i)9-s + (−0.0500 − 1.42i)10-s + (−0.00410 − 0.00831i)11-s + (−0.0782 − 0.996i)12-s + (0.191 + 0.565i)13-s + (0.105 + 0.111i)14-s + (−0.800 − 1.17i)15-s + (−0.920 + 0.391i)16-s + (−0.452 − 0.452i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.590933 + 0.375618i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.590933 + 0.375618i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.09 + 0.894i)T \) |
| 3 | \( 1 + (1.71 + 0.212i)T \) |
| good | 5 | \( 1 + (-2.09 - 2.39i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (0.401 + 0.0529i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (0.0136 + 0.0275i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (-0.691 - 2.03i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (1.86 + 1.86i)T + 17iT^{2} \) |
| 19 | \( 1 + (0.587 - 2.95i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.0505 - 0.384i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (-3.58 + 0.234i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (7.70 - 4.44i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.68 - 8.45i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-0.682 - 5.18i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-0.855 + 0.421i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (-7.12 + 1.90i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.23 + 7.83i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-0.405 - 0.462i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (-0.837 - 12.7i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-2.79 - 1.37i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-4.00 - 9.67i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (2.06 - 4.99i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-2.38 - 8.88i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-0.647 - 0.567i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (12.1 - 5.03i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (3.04 + 1.75i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.84620579733590897286594058456, −10.08233177622995510074153655230, −9.574828653659371329866308151267, −8.343931494870330114644829847252, −6.97337526239257262975483054496, −6.70220134509421448623809079422, −5.54419021494360980754514145835, −4.09172578528894803329992085585, −2.71059892078478157395025944306, −1.52886936937243315272430909704,
0.60150443992762588009988979917, 1.93530070551767598529487372502, 4.38670477881660303939452189913, 5.40223054974116792563928496989, 5.88862012490822596756967920455, 6.83425564545059875434184276685, 7.920025662256191520628655248161, 9.119398693189664202649949689265, 9.423470845922373961110182928745, 10.56596999724733465223782952229
