L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s + 4·11-s + 2·13-s + 16-s − 17-s + 2·20-s + 4·22-s − 25-s + 2·26-s − 10·29-s − 8·31-s + 32-s − 34-s + 2·37-s + 2·40-s + 10·41-s + 12·43-s + 4·44-s − 7·49-s − 50-s + 2·52-s + 6·53-s + 8·55-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 1.20·11-s + 0.554·13-s + 1/4·16-s − 0.242·17-s + 0.447·20-s + 0.852·22-s − 1/5·25-s + 0.392·26-s − 1.85·29-s − 1.43·31-s + 0.176·32-s − 0.171·34-s + 0.328·37-s + 0.316·40-s + 1.56·41-s + 1.82·43-s + 0.603·44-s − 49-s − 0.141·50-s + 0.277·52-s + 0.824·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(6.271620138\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.271620138\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.68802067007920, −13.13039568537319, −12.74256571542795, −12.46079983263071, −11.55194687767298, −11.30681232417555, −10.92568232414291, −10.30045165354897, −9.572392305131629, −9.279338043219121, −8.956923662646919, −8.141124922054058, −7.462754179058109, −7.136288068494579, −6.392184343987051, −5.981580777989327, −5.683280699831704, −5.064896537202601, −4.302377606787809, −3.752361546059185, −3.536038862029324, −2.422413043633208, −2.088224351291452, −1.438059030503147, −0.6843885735996437,
0.6843885735996437, 1.438059030503147, 2.088224351291452, 2.422413043633208, 3.536038862029324, 3.752361546059185, 4.302377606787809, 5.064896537202601, 5.683280699831704, 5.981580777989327, 6.392184343987051, 7.136288068494579, 7.462754179058109, 8.141124922054058, 8.956923662646919, 9.279338043219121, 9.572392305131629, 10.30045165354897, 10.92568232414291, 11.30681232417555, 11.55194687767298, 12.46079983263071, 12.74256571542795, 13.13039568537319, 13.68802067007920