L(s) = 1 | − 2.17i·2-s + 3-s − 2.70·4-s + 0.630i·5-s − 2.17i·6-s − i·7-s + 1.53i·8-s + 9-s + 1.36·10-s − 5.70i·11-s − 2.70·12-s + (−2.17 − 2.87i)13-s − 2.17·14-s + 0.630i·15-s − 2.07·16-s + 1.07·17-s + ⋯ |
L(s) = 1 | − 1.53i·2-s + 0.577·3-s − 1.35·4-s + 0.282i·5-s − 0.885i·6-s − 0.377i·7-s + 0.544i·8-s + 0.333·9-s + 0.432·10-s − 1.72i·11-s − 0.782·12-s + (−0.601 − 0.798i)13-s − 0.579·14-s + 0.162i·15-s − 0.519·16-s + 0.261·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.798 + 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.451937 - 1.35053i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.451937 - 1.35053i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (2.17 + 2.87i)T \) |
good | 2 | \( 1 + 2.17iT - 2T^{2} \) |
| 5 | \( 1 - 0.630iT - 5T^{2} \) |
| 11 | \( 1 + 5.70iT - 11T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 - 7.41iT - 19T^{2} \) |
| 23 | \( 1 - 5.41T + 23T^{2} \) |
| 29 | \( 1 - 6.68T + 29T^{2} \) |
| 31 | \( 1 - 6.15iT - 31T^{2} \) |
| 37 | \( 1 - 3.41iT - 37T^{2} \) |
| 41 | \( 1 - 1.21iT - 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 6.04iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 1.36iT - 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 - 0.581iT - 67T^{2} \) |
| 71 | \( 1 + 4.81iT - 71T^{2} \) |
| 73 | \( 1 + 3.81iT - 73T^{2} \) |
| 79 | \( 1 + 1.65T + 79T^{2} \) |
| 83 | \( 1 + 12.8iT - 83T^{2} \) |
| 89 | \( 1 - 6.20iT - 89T^{2} \) |
| 97 | \( 1 + 1.07iT - 97T^{2} \) |
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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
−11.43432955357083461865826726520, −10.50058292325122204817204286754, −10.08657142583933243525912579529, −8.811975164797798970238473440675, −8.034362462366570949678226393236, −6.56506880454803847528069253209, −4.98543438906148982988847618257, −3.41077632709286330860510459863, −3.00785803725823986099478664975, −1.16562027456661869061557140713,
2.40523797121673712154553993532, 4.55756201184173832506471217065, 5.07896967136810093548979957172, 6.87077313329531232272053978931, 7.05065626201197100117786875001, 8.336436245641758645490335419890, 9.130046766844902550691042705760, 9.839564182423097390096652620144, 11.46837167978456605611003492691, 12.60025084524826471231356665192