L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 2·13-s + 14-s + 16-s − 17-s − 6·19-s + 8·23-s − 5·25-s + 2·26-s + 28-s − 6·29-s − 8·31-s + 32-s − 34-s + 6·37-s − 6·38-s − 6·41-s + 2·43-s + 8·46-s − 2·47-s + 49-s − 5·50-s + 2·52-s + 2·53-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s − 1.37·19-s + 1.66·23-s − 25-s + 0.392·26-s + 0.188·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 0.171·34-s + 0.986·37-s − 0.973·38-s − 0.937·41-s + 0.304·43-s + 1.17·46-s − 0.291·47-s + 1/7·49-s − 0.707·50-s + 0.277·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−13.08507883978569, −12.70600217642495, −12.27545963416021, −11.51664505128328, −11.24697838045176, −10.90259640576471, −10.54690277256119, −9.770498038888355, −9.361268241408455, −8.802281112650262, −8.411013606131987, −7.795999031289682, −7.384563900326118, −6.814689483193828, −6.381587921466367, −5.892386868044911, −5.281516339321981, −4.992238784483301, −4.295564885582373, −3.789059906683940, −3.500639940061246, −2.662652138763074, −2.105421522956333, −1.680318100415121, −0.8834900617934954, 0,
0.8834900617934954, 1.680318100415121, 2.105421522956333, 2.662652138763074, 3.500639940061246, 3.789059906683940, 4.295564885582373, 4.992238784483301, 5.281516339321981, 5.892386868044911, 6.381587921466367, 6.814689483193828, 7.384563900326118, 7.795999031289682, 8.411013606131987, 8.802281112650262, 9.361268241408455, 9.770498038888355, 10.54690277256119, 10.90259640576471, 11.24697838045176, 11.51664505128328, 12.27545963416021, 12.70600217642495, 13.08507883978569