L(s) = 1 | − 3·9-s + 4·11-s − 6·13-s + 6·17-s + 19-s − 8·23-s + 2·29-s + 2·37-s + 2·41-s + 4·43-s + 8·47-s − 7·49-s − 6·53-s + 4·59-s + 2·61-s + 8·67-s + 8·71-s − 2·73-s − 8·79-s + 9·81-s + 4·83-s − 14·89-s − 14·97-s − 12·99-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 9-s + 1.20·11-s − 1.66·13-s + 1.45·17-s + 0.229·19-s − 1.66·23-s + 0.371·29-s + 0.328·37-s + 0.312·41-s + 0.609·43-s + 1.16·47-s − 49-s − 0.824·53-s + 0.520·59-s + 0.256·61-s + 0.977·67-s + 0.949·71-s − 0.234·73-s − 0.900·79-s + 81-s + 0.439·83-s − 1.48·89-s − 1.42·97-s − 1.20·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.645203992\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.645203992\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−14.91286620618087, −14.39626570027398, −14.23064332337009, −13.87619725042561, −12.80332619117240, −12.33463163819719, −11.94014750178283, −11.59750839661701, −10.89264642055083, −10.12679593151882, −9.659125449896059, −9.376277633059672, −8.525769082572116, −7.971639379003492, −7.547892378004358, −6.813061851508708, −6.193533049350993, −5.599337511036859, −5.124605155159121, −4.255930295235811, −3.734614218845040, −2.907999984616895, −2.363701515068319, −1.453210936848285, −0.4964548518544728,
0.4964548518544728, 1.453210936848285, 2.363701515068319, 2.907999984616895, 3.734614218845040, 4.255930295235811, 5.124605155159121, 5.599337511036859, 6.193533049350993, 6.813061851508708, 7.547892378004358, 7.971639379003492, 8.525769082572116, 9.376277633059672, 9.659125449896059, 10.12679593151882, 10.89264642055083, 11.59750839661701, 11.94014750178283, 12.33463163819719, 12.80332619117240, 13.87619725042561, 14.23064332337009, 14.39626570027398, 14.91286620618087