L(s) = 1 | + (0.841 − 0.540i)2-s + (0.142 − 0.989i)3-s + (0.415 − 0.909i)4-s + (−1.72 + 0.505i)5-s + (−0.415 − 0.909i)6-s + (−0.654 − 0.755i)7-s + (−0.142 − 0.989i)8-s + (−0.959 − 0.281i)9-s + (−1.17 + 1.35i)10-s + (−2.35 − 1.51i)11-s + (−0.841 − 0.540i)12-s + (−0.0968 + 0.111i)13-s + (−0.959 − 0.281i)14-s + (0.255 + 1.77i)15-s + (−0.654 − 0.755i)16-s + (−1.12 − 2.46i)17-s + ⋯ |
L(s) = 1 | + (0.594 − 0.382i)2-s + (0.0821 − 0.571i)3-s + (0.207 − 0.454i)4-s + (−0.769 + 0.226i)5-s + (−0.169 − 0.371i)6-s + (−0.247 − 0.285i)7-s + (−0.0503 − 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.371 + 0.428i)10-s + (−0.710 − 0.456i)11-s + (−0.242 − 0.156i)12-s + (−0.0268 + 0.0309i)13-s + (−0.256 − 0.0752i)14-s + (0.0659 + 0.458i)15-s + (−0.163 − 0.188i)16-s + (−0.272 − 0.597i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0990461 + 0.808217i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0990461 + 0.808217i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.841 + 0.540i)T \) |
| 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 7 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (3.42 + 3.35i)T \) |
good | 5 | \( 1 + (1.72 - 0.505i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (2.35 + 1.51i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.0968 - 0.111i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.12 + 2.46i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (1.42 - 3.12i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.588 + 1.28i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.296 - 2.05i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (7.06 + 2.07i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-0.863 + 0.253i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.00386 + 0.0268i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 1.43T + 47T^{2} \) |
| 53 | \( 1 + (5.62 + 6.48i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-4.94 + 5.70i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.620 - 4.31i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-4.60 + 2.96i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-5.51 + 3.54i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.364 + 0.797i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-1.94 + 2.24i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (4.34 + 1.27i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.345 + 2.40i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.39 + 0.702i)T + (81.6 - 52.4i)T^{2} \) |
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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
−9.767143693056909668666607478626, −8.557883608476418010177242834405, −7.81652730794303447194609190427, −6.99578432382244928872137247125, −6.13722905506048331779098680508, −5.13827393896538324842641699756, −3.99267964690772393063880750812, −3.18314645508426106870195754300, −2.04188230509789387345828437869, −0.28417626254980046804154503114,
2.30666614634914331363875030830, 3.50696892819545202484623015358, 4.30591134091273218934089058789, 5.13589603014214619522248548870, 6.04327640527587882176974814769, 7.11005912811351512918955214286, 7.966820648002199133995214296696, 8.633359300801063276665561391776, 9.617627191535423379240165617230, 10.51010105091994203224439940134