L(s) = 1 | + 2-s − i·3-s + 4-s + (2.23 + 0.0950i)5-s − i·6-s − 3.20i·7-s + 8-s − 9-s + (2.23 + 0.0950i)10-s − 3.73·11-s − i·12-s − 0.424·13-s − 3.20i·14-s + (0.0950 − 2.23i)15-s + 16-s + 6.90·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577i·3-s + 0.5·4-s + (0.999 + 0.0425i)5-s − 0.408i·6-s − 1.21i·7-s + 0.353·8-s − 0.333·9-s + (0.706 + 0.0300i)10-s − 1.12·11-s − 0.288i·12-s − 0.117·13-s − 0.857i·14-s + (0.0245 − 0.576i)15-s + 0.250·16-s + 1.67·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.197 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.787650296\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.787650296\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-2.23 - 0.0950i)T \) |
| 37 | \( 1 + (5.90 - 1.45i)T \) |
good | 7 | \( 1 + 3.20iT - 7T^{2} \) |
| 11 | \( 1 + 3.73T + 11T^{2} \) |
| 13 | \( 1 + 0.424T + 13T^{2} \) |
| 17 | \( 1 - 6.90T + 17T^{2} \) |
| 19 | \( 1 + 8.02iT - 19T^{2} \) |
| 23 | \( 1 + 6.34T + 23T^{2} \) |
| 29 | \( 1 - 3.39iT - 29T^{2} \) |
| 31 | \( 1 - 1.20iT - 31T^{2} \) |
| 41 | \( 1 - 5.74T + 41T^{2} \) |
| 43 | \( 1 - 9.31T + 43T^{2} \) |
| 47 | \( 1 + 2.04iT - 47T^{2} \) |
| 53 | \( 1 - 11.8iT - 53T^{2} \) |
| 59 | \( 1 + 2.16iT - 59T^{2} \) |
| 61 | \( 1 + 1.45iT - 61T^{2} \) |
| 67 | \( 1 - 11.4iT - 67T^{2} \) |
| 71 | \( 1 + 5.59T + 71T^{2} \) |
| 73 | \( 1 + 12.5iT - 73T^{2} \) |
| 79 | \( 1 - 4.50iT - 79T^{2} \) |
| 83 | \( 1 - 7.00iT - 83T^{2} \) |
| 89 | \( 1 + 15.9iT - 89T^{2} \) |
| 97 | \( 1 + 4.43T + 97T^{2} \) |
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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
−9.968249828022633912021874600778, −8.808130305919845716926059020918, −7.53637367148972662064724944691, −7.28839284527787780099637047078, −6.16737590129686190834192827402, −5.43994289269224574581383692770, −4.57834873309220474100974464194, −3.25046417249713860294226705476, −2.34766319110349515709494804142, −0.998358224457116856124306351672,
1.91455134089911336377150820316, 2.77561047447596708743885010955, 3.83447314140033150360117673805, 5.14946884928254818145702863991, 5.74239902488945881566903549464, 6.02962629626710722979384610825, 7.67208684333597658153120940514, 8.302606149661874363907055787227, 9.465026076721944112104799459279, 10.07878750648590874397215187549