| L(s) = 1 | − 4·5-s + 7·7-s − 26·11-s + 2·13-s + 36·17-s + 76·19-s − 114·23-s − 109·25-s − 6·29-s + 256·31-s − 28·35-s − 86·37-s − 160·41-s + 220·43-s + 308·47-s + 49·49-s − 258·53-s + 104·55-s + 264·59-s + 606·61-s − 8·65-s + 520·67-s − 286·71-s − 530·73-s − 182·77-s + 44·79-s + 1.01e3·83-s + ⋯ |
| L(s) = 1 | − 0.357·5-s + 0.377·7-s − 0.712·11-s + 0.0426·13-s + 0.513·17-s + 0.917·19-s − 1.03·23-s − 0.871·25-s − 0.0384·29-s + 1.48·31-s − 0.135·35-s − 0.382·37-s − 0.609·41-s + 0.780·43-s + 0.955·47-s + 1/7·49-s − 0.668·53-s + 0.254·55-s + 0.582·59-s + 1.27·61-s − 0.0152·65-s + 0.948·67-s − 0.478·71-s − 0.849·73-s − 0.269·77-s + 0.0626·79-s + 1.33·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.802547320\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.802547320\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 + 26 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 36 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 114 T + p^{3} T^{2} \) |
| 29 | \( 1 + 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 256 T + p^{3} T^{2} \) |
| 37 | \( 1 + 86 T + p^{3} T^{2} \) |
| 41 | \( 1 + 160 T + p^{3} T^{2} \) |
| 43 | \( 1 - 220 T + p^{3} T^{2} \) |
| 47 | \( 1 - 308 T + p^{3} T^{2} \) |
| 53 | \( 1 + 258 T + p^{3} T^{2} \) |
| 59 | \( 1 - 264 T + p^{3} T^{2} \) |
| 61 | \( 1 - 606 T + p^{3} T^{2} \) |
| 67 | \( 1 - 520 T + p^{3} T^{2} \) |
| 71 | \( 1 + 286 T + p^{3} T^{2} \) |
| 73 | \( 1 + 530 T + p^{3} T^{2} \) |
| 79 | \( 1 - 44 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1012 T + p^{3} T^{2} \) |
| 89 | \( 1 + 768 T + p^{3} T^{2} \) |
| 97 | \( 1 - 222 T + p^{3} T^{2} \) |
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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
−9.771964356046479408831914258257, −8.576635746814022645550605385583, −7.909892269338559156756092729838, −7.27086165784982874034878305458, −6.06248896791518300164963031036, −5.26919390985920422231073202729, −4.28191529239565793326720288774, −3.26417217377954346050117578041, −2.09989662181864373045519307577, −0.71204894643662531912891323597,
0.71204894643662531912891323597, 2.09989662181864373045519307577, 3.26417217377954346050117578041, 4.28191529239565793326720288774, 5.26919390985920422231073202729, 6.06248896791518300164963031036, 7.27086165784982874034878305458, 7.909892269338559156756092729838, 8.576635746814022645550605385583, 9.771964356046479408831914258257
