L(s) = 1 | − 2·3-s + 9-s − 3·11-s + 2·13-s − 4·17-s − 3·23-s + 4·27-s + 29-s + 2·31-s + 6·33-s + 7·37-s − 4·39-s − 2·41-s − 43-s + 12·47-s + 8·51-s + 6·53-s − 6·59-s − 6·61-s − 7·67-s + 6·69-s − 3·71-s + 2·73-s − 5·79-s − 11·81-s + 6·83-s − 2·87-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.904·11-s + 0.554·13-s − 0.970·17-s − 0.625·23-s + 0.769·27-s + 0.185·29-s + 0.359·31-s + 1.04·33-s + 1.15·37-s − 0.640·39-s − 0.312·41-s − 0.152·43-s + 1.75·47-s + 1.12·51-s + 0.824·53-s − 0.781·59-s − 0.768·61-s − 0.855·67-s + 0.722·69-s − 0.356·71-s + 0.234·73-s − 0.562·79-s − 1.22·81-s + 0.658·83-s − 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−7.29503280395937624322715330374, −6.34183013122515993030036065127, −6.09353415652958882967463984380, −5.32726855896231454791568025930, −4.68373124607300292898041108860, −4.02813466130325722841120252491, −2.94166430898745795507721117378, −2.16361017716306897017219769826, −0.950923882251271936162469910195, 0,
0.950923882251271936162469910195, 2.16361017716306897017219769826, 2.94166430898745795507721117378, 4.02813466130325722841120252491, 4.68373124607300292898041108860, 5.32726855896231454791568025930, 6.09353415652958882967463984380, 6.34183013122515993030036065127, 7.29503280395937624322715330374