L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 12-s − 2·13-s − 14-s + 15-s + 16-s − 6·17-s − 18-s − 20-s − 21-s + 24-s + 25-s + 2·26-s − 27-s + 28-s + 6·29-s − 30-s + 4·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.223·20-s − 0.218·21-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.48268682006495, −13.69150396832400, −13.32764739929048, −12.63346009772679, −12.06449176611455, −11.85580107068486, −11.16914453599721, −10.85607282860597, −10.38909363848803, −9.740287608499211, −9.319370476029118, −8.669647676210446, −8.125260685376139, −7.851682036922090, −6.992299806292799, −6.645463453741115, −6.309387730279863, −5.303644110701208, −4.949369458613944, −4.346578033006387, −3.728642154150526, −2.835376613160811, −2.284159682530498, −1.543125877851479, −0.7274454652320339, 0,
0.7274454652320339, 1.543125877851479, 2.284159682530498, 2.835376613160811, 3.728642154150526, 4.346578033006387, 4.949369458613944, 5.303644110701208, 6.309387730279863, 6.645463453741115, 6.992299806292799, 7.851682036922090, 8.125260685376139, 8.669647676210446, 9.319370476029118, 9.740287608499211, 10.38909363848803, 10.85607282860597, 11.16914453599721, 11.85580107068486, 12.06449176611455, 12.63346009772679, 13.32764739929048, 13.69150396832400, 14.48268682006495