L(s) = 1 | + 3-s − 7-s + 9-s − 5·11-s + 2·13-s + 4·17-s − 6·19-s − 21-s + 23-s + 27-s + 29-s − 2·31-s − 5·33-s + 5·37-s + 2·39-s + 7·43-s + 2·47-s + 49-s + 4·51-s − 6·53-s − 6·57-s + 6·59-s − 63-s − 3·67-s + 69-s − 9·71-s − 12·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 0.554·13-s + 0.970·17-s − 1.37·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s + 0.185·29-s − 0.359·31-s − 0.870·33-s + 0.821·37-s + 0.320·39-s + 1.06·43-s + 0.291·47-s + 1/7·49-s + 0.560·51-s − 0.824·53-s − 0.794·57-s + 0.781·59-s − 0.125·63-s − 0.366·67-s + 0.120·69-s − 1.06·71-s − 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.62261978581837315844844791735, −6.84975420893341449107032481591, −6.00337050173580347036694206133, −5.46775666885415928197583872952, −4.52775310146841278356834318594, −3.84739083688583317756909361917, −2.92346284479968560239805776911, −2.45687017503626336479555663646, −1.31636760612629491864760333653, 0,
1.31636760612629491864760333653, 2.45687017503626336479555663646, 2.92346284479968560239805776911, 3.84739083688583317756909361917, 4.52775310146841278356834318594, 5.46775666885415928197583872952, 6.00337050173580347036694206133, 6.84975420893341449107032481591, 7.62261978581837315844844791735