L(s) = 1 | + 2·5-s − 6·11-s + 6·13-s − 2·17-s − 4·19-s − 2·23-s − 25-s − 8·29-s − 4·31-s − 6·37-s + 10·41-s − 4·43-s − 4·47-s + 4·53-s − 12·55-s − 12·59-s + 2·61-s + 12·65-s + 12·67-s − 6·71-s + 2·73-s − 8·79-s − 4·85-s − 14·89-s − 8·95-s + 2·97-s − 18·101-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.80·11-s + 1.66·13-s − 0.485·17-s − 0.917·19-s − 0.417·23-s − 1/5·25-s − 1.48·29-s − 0.718·31-s − 0.986·37-s + 1.56·41-s − 0.609·43-s − 0.583·47-s + 0.549·53-s − 1.61·55-s − 1.56·59-s + 0.256·61-s + 1.48·65-s + 1.46·67-s − 0.712·71-s + 0.234·73-s − 0.900·79-s − 0.433·85-s − 1.48·89-s − 0.820·95-s + 0.203·97-s − 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−8.256281311226866294496813291431, −7.54997069645197216277073706475, −6.58533741844578296970363838022, −5.78343607874588563482284182474, −5.46524989415668552363508752802, −4.33717318093833859660215502627, −3.44124710132985509655416514471, −2.36826211594438163202639427032, −1.66994087876955429718916737640, 0,
1.66994087876955429718916737640, 2.36826211594438163202639427032, 3.44124710132985509655416514471, 4.33717318093833859660215502627, 5.46524989415668552363508752802, 5.78343607874588563482284182474, 6.58533741844578296970363838022, 7.54997069645197216277073706475, 8.256281311226866294496813291431