L(s) = 1 | + 2-s + 4-s + 8-s + 11-s + 4·13-s + 16-s − 6·17-s + 4·19-s + 22-s − 6·23-s − 5·25-s + 4·26-s − 6·29-s − 8·31-s + 32-s − 6·34-s − 10·37-s + 4·38-s + 6·41-s + 8·43-s + 44-s − 6·46-s − 6·47-s − 5·50-s + 4·52-s − 6·58-s − 8·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 0.301·11-s + 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.213·22-s − 1.25·23-s − 25-s + 0.784·26-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s − 1.64·37-s + 0.648·38-s + 0.937·41-s + 1.21·43-s + 0.150·44-s − 0.884·46-s − 0.875·47-s − 0.707·50-s + 0.554·52-s − 0.787·58-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.29087986975847329500837910780, −6.52504829438381665573881626069, −5.85955462295020247251479297795, −5.44687910527563212016596341412, −4.36376964393171453387912321319, −3.89761457992935995656155838511, −3.25261158060160886155862976042, −2.12020721928430735862314078108, −1.54124369917794574272131224237, 0,
1.54124369917794574272131224237, 2.12020721928430735862314078108, 3.25261158060160886155862976042, 3.89761457992935995656155838511, 4.36376964393171453387912321319, 5.44687910527563212016596341412, 5.85955462295020247251479297795, 6.52504829438381665573881626069, 7.29087986975847329500837910780