L(s) = 1 | − 2.84·2-s + 2.18·3-s + 4.10·4-s − 6.21·6-s − 9.15i·7-s − 0.287·8-s − 4.22·9-s + 9.89·11-s + 8.95·12-s + 12.3·13-s + 26.0i·14-s − 15.5·16-s − 11.1i·17-s + 12.0·18-s + (−18.7 + 2.97i)19-s + ⋯ |
L(s) = 1 | − 1.42·2-s + 0.728·3-s + 1.02·4-s − 1.03·6-s − 1.30i·7-s − 0.0359·8-s − 0.469·9-s + 0.899·11-s + 0.746·12-s + 0.948·13-s + 1.86i·14-s − 0.974·16-s − 0.658i·17-s + 0.668·18-s + (−0.987 + 0.156i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.301 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8196395140\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8196395140\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (18.7 - 2.97i)T \) |
good | 2 | \( 1 + 2.84T + 4T^{2} \) |
| 3 | \( 1 - 2.18T + 9T^{2} \) |
| 7 | \( 1 + 9.15iT - 49T^{2} \) |
| 11 | \( 1 - 9.89T + 121T^{2} \) |
| 13 | \( 1 - 12.3T + 169T^{2} \) |
| 17 | \( 1 + 11.1iT - 289T^{2} \) |
| 23 | \( 1 + 20.8iT - 529T^{2} \) |
| 29 | \( 1 + 10.5iT - 841T^{2} \) |
| 31 | \( 1 - 52.3iT - 961T^{2} \) |
| 37 | \( 1 - 10.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 42.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 45.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 4.22iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 95.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + 6.38iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 14.8T + 3.72e3T^{2} \) |
| 67 | \( 1 - 38.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 18.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 137. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 127. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 19.8iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 42.4T + 9.40e3T^{2} \) |
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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
−10.53404709934552236799060936805, −9.383206959667678695801520558575, −8.764618763230181674535350535597, −8.107096599789155750128219354450, −7.15553566683789707847268556042, −6.36550869391668151042721703892, −4.47638623062222029159387457504, −3.40759186251035256686102561650, −1.80555596463279502603861215820, −0.50790796565402952233311626300,
1.52878251262222705482365866014, 2.61770749259462525814920608493, 4.02653113044073150659750740011, 5.76777083702303122043323658739, 6.60645537503761079109669829935, 8.157352686418561531184676358913, 8.309592104778055945806432019922, 9.292751065256741327129163333075, 9.598958943580584398450321197244, 11.12375906982738708528239112267