L(s) = 1 | − 2·3-s + 9-s + 4·11-s + 2·13-s + 3·17-s + 3·23-s + 4·27-s − 6·29-s − 9·31-s − 8·33-s − 4·39-s − 5·41-s − 6·43-s − 9·47-s − 6·51-s − 6·53-s − 8·59-s − 8·61-s + 14·67-s − 6·69-s + 11·71-s + 2·73-s + 9·79-s − 11·81-s + 6·83-s + 12·87-s − 11·89-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.727·17-s + 0.625·23-s + 0.769·27-s − 1.11·29-s − 1.61·31-s − 1.39·33-s − 0.640·39-s − 0.780·41-s − 0.914·43-s − 1.31·47-s − 0.840·51-s − 0.824·53-s − 1.04·59-s − 1.02·61-s + 1.71·67-s − 0.722·69-s + 1.30·71-s + 0.234·73-s + 1.01·79-s − 1.22·81-s + 0.658·83-s + 1.28·87-s − 1.16·89-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 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−7.11044115657832784025917734940, −6.53044058171096431969337711478, −6.02294350817926068884566442199, −5.29081621388565372134995285742, −4.80573565140868679309569618374, −3.70913989162551534996288906458, −3.32847973219145791341358740365, −1.86197584730658228901550371371, −1.13880264951000867601933654827, 0,
1.13880264951000867601933654827, 1.86197584730658228901550371371, 3.32847973219145791341358740365, 3.70913989162551534996288906458, 4.80573565140868679309569618374, 5.29081621388565372134995285742, 6.02294350817926068884566442199, 6.53044058171096431969337711478, 7.11044115657832784025917734940