L(s) = 1 | − 2·3-s − 5-s + 9-s + 2·13-s + 2·15-s − 6·17-s − 2·23-s + 25-s + 4·27-s + 8·29-s + 4·31-s − 2·37-s − 4·39-s + 4·41-s + 4·43-s − 45-s + 2·47-s − 7·49-s + 12·51-s − 10·53-s − 8·61-s − 2·65-s − 2·67-s + 4·69-s + 8·71-s + 10·73-s − 2·75-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s + 0.554·13-s + 0.516·15-s − 1.45·17-s − 0.417·23-s + 1/5·25-s + 0.769·27-s + 1.48·29-s + 0.718·31-s − 0.328·37-s − 0.640·39-s + 0.624·41-s + 0.609·43-s − 0.149·45-s + 0.291·47-s − 49-s + 1.68·51-s − 1.37·53-s − 1.02·61-s − 0.248·65-s − 0.244·67-s + 0.481·69-s + 0.949·71-s + 1.17·73-s − 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.973369609328016739802260600927, −6.94502565858199251934678330245, −6.38222005346185640960055390825, −5.90381201296459453545708531275, −4.79235296487685314199498668214, −4.50230140225685107342490263314, −3.40327548932625686367966986952, −2.38216406968926952108746325867, −1.06829173800301736083652189295, 0,
1.06829173800301736083652189295, 2.38216406968926952108746325867, 3.40327548932625686367966986952, 4.50230140225685107342490263314, 4.79235296487685314199498668214, 5.90381201296459453545708531275, 6.38222005346185640960055390825, 6.94502565858199251934678330245, 7.973369609328016739802260600927