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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3020936297\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3020936297\) |
\(L(\frac{3}{2})\) |
|
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
−9.414649604601669438310470580675, −8.823870762994090661242419414014, −8.362367273170359784737396709188, −7.27139926452396677185915392440, −6.52077852440277858066546832098, −5.20108730770151976255151072944, −4.61654395462939448843453810250, −3.09225472628970756869268564914, −2.43346613747962732211237618972, −0.19431395075347791987408952303,
1.07368554555471315914285004436, 2.56937586563418817682063556446, 3.76833216999829774679003071241, 4.71096534885493830374000040041, 6.08867155112195702470790604333, 7.02249330634746044048974172356, 7.63778061793114876802823376007, 8.146201157612888196975945891093, 9.003639554745667505053729124283, 10.18985496673773360101505345876