L(s) = 1 | − 3-s + 2·7-s + 9-s − 6·11-s − 2·13-s + 6·17-s + 4·19-s − 2·21-s − 8·23-s − 27-s + 8·31-s + 6·33-s + 2·37-s + 2·39-s − 6·41-s − 4·43-s − 4·47-s − 3·49-s − 6·51-s + 6·53-s − 4·57-s − 6·59-s + 6·61-s + 2·63-s + 8·69-s + 4·71-s − 12·73-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.80·11-s − 0.554·13-s + 1.45·17-s + 0.917·19-s − 0.436·21-s − 1.66·23-s − 0.192·27-s + 1.43·31-s + 1.04·33-s + 0.328·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.583·47-s − 3/7·49-s − 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.781·59-s + 0.768·61-s + 0.251·63-s + 0.963·69-s + 0.474·71-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−7.998226557785853336531874963491, −7.39112471498491388202019710037, −6.38749811340339155752593707250, −5.42617391602824052281485698474, −5.23051054059967605845738609486, −4.38033331724880641091637291446, −3.25267604463798619981167116936, −2.39503419123733242408909442443, −1.29979159080831887070426971430, 0,
1.29979159080831887070426971430, 2.39503419123733242408909442443, 3.25267604463798619981167116936, 4.38033331724880641091637291446, 5.23051054059967605845738609486, 5.42617391602824052281485698474, 6.38749811340339155752593707250, 7.39112471498491388202019710037, 7.998226557785853336531874963491