| 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
−10.51010105091994203224439940134, −9.617627191535423379240165617230, −8.633359300801063276665561391776, −7.966820648002199133995214296696, −7.11005912811351512918955214286, −6.04327640527587882176974814769, −5.13589603014214619522248548870, −4.30591134091273218934089058789, −3.50696892819545202484623015358, −2.30666614634914331363875030830,
0.28417626254980046804154503114, 2.04188230509789387345828437869, 3.18314645508426106870195754300, 3.99267964690772393063880750812, 5.13827393896538324842641699756, 6.13722905506048331779098680508, 6.99578432382244928872137247125, 7.81652730794303447194609190427, 8.557883608476418010177242834405, 9.767143693056909668666607478626
