L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 2·10-s − 2·11-s + 2·13-s + 16-s − 6·17-s − 4·19-s − 2·20-s − 2·22-s − 23-s − 25-s + 2·26-s − 2·29-s − 2·31-s + 32-s − 6·34-s − 4·38-s − 2·40-s + 10·43-s − 2·44-s − 46-s − 2·47-s − 50-s + 2·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 0.632·10-s − 0.603·11-s + 0.554·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 0.447·20-s − 0.426·22-s − 0.208·23-s − 1/5·25-s + 0.392·26-s − 0.371·29-s − 0.359·31-s + 0.176·32-s − 1.02·34-s − 0.648·38-s − 0.316·40-s + 1.52·43-s − 0.301·44-s − 0.147·46-s − 0.291·47-s − 0.141·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20286 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20286 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.703176572\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.703176572\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−15.53947289947982, −15.18576424381841, −14.68449295226602, −13.87800320663152, −13.47443988768051, −12.90377542569563, −12.46749408718791, −11.82507200031862, −11.23887811570246, −10.82644670601387, −10.42612043269258, −9.422954454917944, −8.842038143630845, −8.231316336817906, −7.651962794149550, −7.121917224879832, −6.357867677630357, −5.921456071763029, −5.080228012891286, −4.352340962158448, −4.038685474571053, −3.272722816422360, −2.443009301623202, −1.795966688158687, −0.4561144471777923,
0.4561144471777923, 1.795966688158687, 2.443009301623202, 3.272722816422360, 4.038685474571053, 4.352340962158448, 5.080228012891286, 5.921456071763029, 6.357867677630357, 7.121917224879832, 7.651962794149550, 8.231316336817906, 8.842038143630845, 9.422954454917944, 10.42612043269258, 10.82644670601387, 11.23887811570246, 11.82507200031862, 12.46749408718791, 12.90377542569563, 13.47443988768051, 13.87800320663152, 14.68449295226602, 15.18576424381841, 15.53947289947982