L(s) = 1 | − 3-s − 7-s + 9-s − 3·11-s + 2·13-s + 17-s − 19-s + 21-s − 2·23-s − 27-s + 5·29-s − 6·31-s + 3·33-s − 37-s − 2·39-s + 5·41-s + 4·43-s + 13·47-s − 6·49-s − 51-s − 11·53-s + 57-s − 4·59-s − 2·61-s − 63-s + 8·67-s + 2·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.554·13-s + 0.242·17-s − 0.229·19-s + 0.218·21-s − 0.417·23-s − 0.192·27-s + 0.928·29-s − 1.07·31-s + 0.522·33-s − 0.164·37-s − 0.320·39-s + 0.780·41-s + 0.609·43-s + 1.89·47-s − 6/7·49-s − 0.140·51-s − 1.51·53-s + 0.132·57-s − 0.520·59-s − 0.256·61-s − 0.125·63-s + 0.977·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−15.94174575382459, −15.61539454192281, −14.89430061068440, −14.10441887292806, −13.85007221896921, −12.88712991924200, −12.76499792206211, −12.17709347761909, −11.40959739378503, −10.93349121956892, −10.43980408219051, −9.940014342944272, −9.219848318357169, −8.687304725801129, −7.856316671766557, −7.504353637826127, −6.689269355841410, −6.084637920218701, −5.641471537266775, −4.917440231835465, −4.262039860227353, −3.513654094185229, −2.771977684208900, −1.946208774114015, −0.9486062414980289, 0,
0.9486062414980289, 1.946208774114015, 2.771977684208900, 3.513654094185229, 4.262039860227353, 4.917440231835465, 5.641471537266775, 6.084637920218701, 6.689269355841410, 7.504353637826127, 7.856316671766557, 8.687304725801129, 9.219848318357169, 9.940014342944272, 10.43980408219051, 10.93349121956892, 11.40959739378503, 12.17709347761909, 12.76499792206211, 12.88712991924200, 13.85007221896921, 14.10441887292806, 14.89430061068440, 15.61539454192281, 15.94174575382459