L(s) = 1 | + (−0.736 − 0.676i)2-s + (−0.809 + 0.587i)3-s + (0.0855 + 0.996i)4-s + (−0.0556 − 1.94i)5-s + (0.993 + 0.113i)6-s + (−0.288 + 0.546i)7-s + (0.610 − 0.791i)8-s + (0.309 − 0.951i)9-s + (−1.27 + 1.47i)10-s + (−0.654 + 0.755i)11-s + (−0.654 − 0.755i)12-s + (0.582 − 0.207i)14-s + (1.18 + 1.54i)15-s + (−0.985 + 0.170i)16-s + (−0.870 + 0.491i)18-s + ⋯ |
L(s) = 1 | + (−0.736 − 0.676i)2-s + (−0.809 + 0.587i)3-s + (0.0855 + 0.996i)4-s + (−0.0556 − 1.94i)5-s + (0.993 + 0.113i)6-s + (−0.288 + 0.546i)7-s + (0.610 − 0.791i)8-s + (0.309 − 0.951i)9-s + (−1.27 + 1.47i)10-s + (−0.654 + 0.755i)11-s + (−0.654 − 0.755i)12-s + (0.582 − 0.207i)14-s + (1.18 + 1.54i)15-s + (−0.985 + 0.170i)16-s + (−0.870 + 0.491i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.06444098151\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06444098151\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.736 + 0.676i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 + (0.654 - 0.755i)T \) |
good | 5 | \( 1 + (0.0556 + 1.94i)T + (-0.998 + 0.0570i)T^{2} \) |
| 7 | \( 1 + (0.288 - 0.546i)T + (-0.564 - 0.825i)T^{2} \) |
| 13 | \( 1 + (-0.696 + 0.717i)T^{2} \) |
| 17 | \( 1 + (0.870 + 0.491i)T^{2} \) |
| 19 | \( 1 + (0.736 + 0.676i)T^{2} \) |
| 23 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 29 | \( 1 + (1.88 + 0.107i)T + (0.993 + 0.113i)T^{2} \) |
| 31 | \( 1 + (0.582 + 0.966i)T + (-0.466 + 0.884i)T^{2} \) |
| 37 | \( 1 + (-0.897 + 0.441i)T^{2} \) |
| 41 | \( 1 + (0.921 - 0.389i)T^{2} \) |
| 43 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 47 | \( 1 + (-0.516 - 0.856i)T^{2} \) |
| 53 | \( 1 + (-1.94 - 0.336i)T + (0.941 + 0.336i)T^{2} \) |
| 59 | \( 1 + (0.100 - 0.498i)T + (-0.921 - 0.389i)T^{2} \) |
| 61 | \( 1 + (-0.0855 + 0.996i)T^{2} \) |
| 67 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 71 | \( 1 + (-0.198 - 0.980i)T^{2} \) |
| 73 | \( 1 + (1.39 - 1.43i)T + (-0.0285 - 0.999i)T^{2} \) |
| 79 | \( 1 + (-0.526 + 0.770i)T + (-0.362 - 0.931i)T^{2} \) |
| 83 | \( 1 + (1.56 - 0.768i)T + (0.610 - 0.791i)T^{2} \) |
| 89 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 97 | \( 1 + (0.0113 - 0.397i)T + (-0.998 - 0.0570i)T^{2} \) |
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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
−9.299410275113010313143397977127, −8.768094162579489804596922379673, −7.890269796555495579819625333881, −7.15233413554940014933366518524, −5.75761482815395999724282024138, −5.34367630869397966573427072700, −4.34458706702690348647654680209, −3.88886630813103855686646517258, −2.35376487943324830452300821727, −1.26920372812347937287392529768,
0.05990269301258509192634415295, 1.80336573659554808955140799790, 2.83743921611862158063309993889, 3.92743832066054206737180422064, 5.41791777775014134168197795868, 5.85893077119867511632567960420, 6.70724649983605959784866273289, 7.16607671429922527945637433601, 7.59064567404646697439783626258, 8.455094629430900051015096145004