L(s) = 1 | + 2·7-s − 3·9-s + 4·11-s + 2·13-s − 4·17-s − 19-s − 6·23-s − 6·29-s + 4·31-s + 10·37-s − 10·41-s + 2·43-s − 6·47-s − 3·49-s − 10·53-s + 2·61-s − 6·63-s + 8·67-s − 4·71-s − 4·73-s + 8·77-s − 4·79-s + 9·81-s − 18·83-s − 2·89-s + 4·91-s − 6·97-s + ⋯ |
L(s) = 1 | + 0.755·7-s − 9-s + 1.20·11-s + 0.554·13-s − 0.970·17-s − 0.229·19-s − 1.25·23-s − 1.11·29-s + 0.718·31-s + 1.64·37-s − 1.56·41-s + 0.304·43-s − 0.875·47-s − 3/7·49-s − 1.37·53-s + 0.256·61-s − 0.755·63-s + 0.977·67-s − 0.474·71-s − 0.468·73-s + 0.911·77-s − 0.450·79-s + 81-s − 1.97·83-s − 0.211·89-s + 0.419·91-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.68665228820403359876290102990, −6.62772059505631963917048900122, −6.23396811127272585822003914830, −5.51279996743150687750479870132, −4.57679697548039743691021654187, −4.02735217023794934128784287101, −3.15972315232164195260326249953, −2.13250398217426296654077716826, −1.39232524073865319113395553341, 0,
1.39232524073865319113395553341, 2.13250398217426296654077716826, 3.15972315232164195260326249953, 4.02735217023794934128784287101, 4.57679697548039743691021654187, 5.51279996743150687750479870132, 6.23396811127272585822003914830, 6.62772059505631963917048900122, 7.68665228820403359876290102990