L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 6·11-s − 2·13-s + 14-s + 16-s − 6·17-s + 2·19-s + 6·22-s + 23-s + 2·26-s − 28-s + 6·29-s + 8·31-s − 32-s + 6·34-s − 8·37-s − 2·38-s − 6·41-s − 2·43-s − 6·44-s − 46-s + 49-s − 2·52-s − 12·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.80·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s + 1.27·22-s + 0.208·23-s + 0.392·26-s − 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s − 1.31·37-s − 0.324·38-s − 0.937·41-s − 0.304·43-s − 0.904·44-s − 0.147·46-s + 1/7·49-s − 0.277·52-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.30380797036247, −13.74389929469114, −13.38374171495138, −12.77259285385300, −12.43781605006901, −11.68071882293531, −11.37418390573723, −10.58622234085723, −10.29712781781926, −9.964588853084029, −9.278880108780039, −8.708678742822853, −8.228401579636796, −7.834527576734443, −7.154436596230064, −6.671447314147272, −6.266841287344142, −5.347597101245206, −4.976829177216775, −4.450036283030068, −3.425300201107648, −2.851382526204125, −2.418042301424138, −1.731826745921210, −0.6523371552714270, 0,
0.6523371552714270, 1.731826745921210, 2.418042301424138, 2.851382526204125, 3.425300201107648, 4.450036283030068, 4.976829177216775, 5.347597101245206, 6.266841287344142, 6.671447314147272, 7.154436596230064, 7.834527576734443, 8.228401579636796, 8.708678742822853, 9.278880108780039, 9.964588853084029, 10.29712781781926, 10.58622234085723, 11.37418390573723, 11.68071882293531, 12.43781605006901, 12.77259285385300, 13.38374171495138, 13.74389929469114, 14.30380797036247