L(s) = 1 | + (0.349 − 0.605i)5-s + (−0.229 + 0.132i)11-s + (−1.13 + 0.657i)13-s + (−1.86 + 3.22i)17-s + (0.382 − 0.220i)19-s + (4.29 + 2.48i)23-s + (2.25 + 3.90i)25-s + (0.273 + 0.157i)29-s + 5.60i·31-s + (−0.351 − 0.608i)37-s + (−5.39 − 9.34i)41-s + (3.73 − 6.46i)43-s − 7.00·47-s + (−8.51 − 4.91i)53-s + 0.185i·55-s + ⋯ |
L(s) = 1 | + (0.156 − 0.270i)5-s + (−0.0692 + 0.0399i)11-s + (−0.315 + 0.182i)13-s + (−0.452 + 0.783i)17-s + (0.0877 − 0.0506i)19-s + (0.896 + 0.517i)23-s + (0.451 + 0.781i)25-s + (0.0507 + 0.0292i)29-s + 1.00i·31-s + (−0.0577 − 0.0999i)37-s + (−0.842 − 1.45i)41-s + (0.569 − 0.985i)43-s − 1.02·47-s + (−1.17 − 0.675i)53-s + 0.0250i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.041822543\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.041822543\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.349 + 0.605i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.229 - 0.132i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.13 - 0.657i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.86 - 3.22i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.382 + 0.220i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.29 - 2.48i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.273 - 0.157i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.60iT - 31T^{2} \) |
| 37 | \( 1 + (0.351 + 0.608i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.39 + 9.34i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.73 + 6.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 7.00T + 47T^{2} \) |
| 53 | \( 1 + (8.51 + 4.91i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 5.65iT - 61T^{2} \) |
| 67 | \( 1 + 5.94T + 67T^{2} \) |
| 71 | \( 1 - 13.4iT - 71T^{2} \) |
| 73 | \( 1 + (-6.66 - 3.84i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 1.39T + 79T^{2} \) |
| 83 | \( 1 + (3.72 - 6.45i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.59 - 9.68i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.18 - 5.30i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.504637027257316545923403772838, −7.68914093227547498385927946941, −6.94240298533649839126532466150, −6.39056521502241479377225603947, −5.29126345475122965422802326337, −4.99996998404339517703572231860, −3.90267219962833604303805236255, −3.19422529218673002013806984566, −2.10233448379506194882402767434, −1.23354394312995586812898301245,
0.27583486044227342634962154120, 1.57246273198988950742622333312, 2.74117276800578395593084532796, 3.15417288225143997899376162283, 4.61767042999517637796646826675, 4.73530824121986923746501718667, 5.99345174946086117830126352668, 6.45196639415625263484398233911, 7.28356711025600328763724188333, 7.906665083846578723622898991124