L(s) = 1 | − 3-s + 5-s + 9-s − 4·11-s − 2·13-s − 15-s − 2·17-s − 4·19-s − 8·23-s + 25-s − 27-s − 6·29-s + 8·31-s + 4·33-s + 2·37-s + 2·39-s − 2·41-s + 12·43-s + 45-s + 8·47-s + 2·51-s − 6·53-s − 4·55-s + 4·57-s + 4·59-s − 2·61-s − 2·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.328·37-s + 0.320·39-s − 0.312·41-s + 1.82·43-s + 0.149·45-s + 1.16·47-s + 0.280·51-s − 0.824·53-s − 0.539·55-s + 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−15.00814555664441, −14.16265139043790, −13.87199732379142, −13.18596958574173, −12.86030464872332, −12.19573516767059, −11.94367801455620, −11.04766013664108, −10.68403604795029, −10.33416080528737, −9.604000666340917, −9.335669805822418, −8.392321648283398, −7.985542912133002, −7.444797063452425, −6.779484318629134, −6.109715687377440, −5.785798089050499, −5.155644197977841, −4.459896024323391, −4.095712592471241, −3.079189419985316, −2.226195309642400, −2.062572889860330, −0.7774050018572963, 0,
0.7774050018572963, 2.062572889860330, 2.226195309642400, 3.079189419985316, 4.095712592471241, 4.459896024323391, 5.155644197977841, 5.785798089050499, 6.109715687377440, 6.779484318629134, 7.444797063452425, 7.985542912133002, 8.392321648283398, 9.335669805822418, 9.604000666340917, 10.33416080528737, 10.68403604795029, 11.04766013664108, 11.94367801455620, 12.19573516767059, 12.86030464872332, 13.18596958574173, 13.87199732379142, 14.16265139043790, 15.00814555664441