| L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s + (−0.553 + 3.13i)5-s + (−0.173 − 0.984i)6-s + (−2.64 + 0.104i)7-s + (−0.500 + 0.866i)8-s + (0.173 + 0.984i)9-s + (−2.44 + 2.04i)10-s + (−1.34 − 2.33i)11-s + (0.500 − 0.866i)12-s + (0.287 + 1.62i)13-s + (−2.09 − 1.61i)14-s + (2.44 − 2.04i)15-s + (−0.939 + 0.342i)16-s + (0.0114 − 0.0651i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (−0.442 − 0.371i)3-s + (0.0868 + 0.492i)4-s + (−0.247 + 1.40i)5-s + (−0.0708 − 0.402i)6-s + (−0.999 + 0.0396i)7-s + (−0.176 + 0.306i)8-s + (0.0578 + 0.328i)9-s + (−0.772 + 0.647i)10-s + (−0.405 − 0.702i)11-s + (0.144 − 0.249i)12-s + (0.0796 + 0.451i)13-s + (−0.559 − 0.432i)14-s + (0.630 − 0.528i)15-s + (−0.234 + 0.0855i)16-s + (0.00278 − 0.0158i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 + 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0934480 - 0.399361i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0934480 - 0.399361i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 7 | \( 1 + (2.64 - 0.104i)T \) |
| 19 | \( 1 + (-0.344 + 4.34i)T \) |
| good | 5 | \( 1 + (0.553 - 3.13i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (1.34 + 2.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.287 - 1.62i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.0114 + 0.0651i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (7.18 + 2.61i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (7.72 + 2.81i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 - 0.361T + 31T^{2} \) |
| 37 | \( 1 + (0.0261 + 0.0452i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.44 - 8.19i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.00422 + 0.00354i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.53 - 8.70i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-1.99 - 11.3i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (0.935 - 5.30i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (5.13 + 1.86i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.75 + 1.47i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (7.69 + 6.45i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (4.08 + 3.42i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (10.4 - 3.81i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (0.339 + 0.587i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.84 + 8.25i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (0.496 - 0.180i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.92162338564018479594120111959, −10.08749588614965082671098382688, −9.019539617305158329255205677940, −7.75383168818522763638137989594, −7.17852334689470923345648725839, −6.19804678335005132734139766649, −5.99218625918628404348247521390, −4.41377438937039779274684682562, −3.31665308026086555934765444415, −2.50174692775854391379562756671,
0.16929094493687319347135527304, 1.82667256997660450105856264491, 3.54996251946861877414943389746, 4.19989230344039057450245567956, 5.36196701379436925524020706200, 5.75953941522856159498856318263, 7.09156568019953368402135908701, 8.182544191862088785390486872915, 9.200363095296480130493062952646, 9.925056909997935655102453690654
