L(s) = 1 | + (−1.34 + 0.450i)2-s + 3-s + (1.59 − 1.20i)4-s + (−0.254 − 2.22i)5-s + (−1.34 + 0.450i)6-s − 2.64i·7-s + (−1.59 + 2.33i)8-s + 9-s + (1.34 + 2.86i)10-s + 1.51i·11-s + (1.59 − 1.20i)12-s + 3.87·13-s + (1.18 + 3.54i)14-s + (−0.254 − 2.22i)15-s + (1.08 − 3.84i)16-s − 3.31i·17-s + ⋯ |
L(s) = 1 | + (−0.947 + 0.318i)2-s + 0.577·3-s + (0.797 − 0.603i)4-s + (−0.113 − 0.993i)5-s + (−0.547 + 0.183i)6-s − 0.998i·7-s + (−0.563 + 0.825i)8-s + 0.333·9-s + (0.423 + 0.905i)10-s + 0.456i·11-s + (0.460 − 0.348i)12-s + 1.07·13-s + (0.317 + 0.946i)14-s + (−0.0656 − 0.573i)15-s + (0.271 − 0.962i)16-s − 0.803i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.821707 - 0.203324i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.821707 - 0.203324i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.34 - 0.450i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (0.254 + 2.22i)T \) |
good | 7 | \( 1 + 2.64iT - 7T^{2} \) |
| 11 | \( 1 - 1.51iT - 11T^{2} \) |
| 13 | \( 1 - 3.87T + 13T^{2} \) |
| 17 | \( 1 + 3.31iT - 17T^{2} \) |
| 19 | \( 1 - 7.08iT - 19T^{2} \) |
| 23 | \( 1 - 4.82iT - 23T^{2} \) |
| 29 | \( 1 + 2.18iT - 29T^{2} \) |
| 31 | \( 1 + 7.36T + 31T^{2} \) |
| 37 | \( 1 + 7.87T + 37T^{2} \) |
| 41 | \( 1 - 8.72T + 41T^{2} \) |
| 43 | \( 1 - 1.01T + 43T^{2} \) |
| 47 | \( 1 - 7.08iT - 47T^{2} \) |
| 53 | \( 1 - 4.50T + 53T^{2} \) |
| 59 | \( 1 + 6.79iT - 59T^{2} \) |
| 61 | \( 1 - 3.60iT - 61T^{2} \) |
| 67 | \( 1 + 1.01T + 67T^{2} \) |
| 71 | \( 1 + 6.72T + 71T^{2} \) |
| 73 | \( 1 - 15.5iT - 73T^{2} \) |
| 79 | \( 1 - 7.36T + 79T^{2} \) |
| 83 | \( 1 - 7.74T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
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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
−13.56015890722135629327704230807, −12.34617353424007395650081583427, −11.09000745901558611758549254850, −9.950859704653270519756366019969, −9.106977496870505737096147149123, −8.040594818726445916504576299185, −7.26016426299637003994358224795, −5.64163550357426164207865405387, −3.89068482881488814454791372069, −1.44147879188092947875719542607,
2.33994703291733375815870286356, 3.52678672159251582304286585508, 6.10440114465324932228124064921, 7.19402210933065850364610019597, 8.534368189674196953509191386461, 9.072787065551751236076089592865, 10.55241982272747408788185000101, 11.14887282898754784888577455960, 12.34882220356501551728628618785, 13.49635622927998105435769137902