| L(s) = 1 | + 74·5-s + 24·7-s + 124·11-s + 478·13-s + 1.19e3·17-s − 3.04e3·19-s + 184·23-s + 2.35e3·25-s + 3.28e3·29-s + 5.72e3·31-s + 1.77e3·35-s + 1.03e4·37-s + 8.88e3·41-s + 9.18e3·43-s + 2.36e4·47-s − 1.62e4·49-s − 1.16e4·53-s + 9.17e3·55-s + 1.68e4·59-s − 1.84e4·61-s + 3.53e4·65-s + 1.55e4·67-s − 3.19e4·71-s − 4.88e3·73-s + 2.97e3·77-s − 4.45e4·79-s + 6.73e4·83-s + ⋯ |
| L(s) = 1 | + 1.32·5-s + 0.185·7-s + 0.308·11-s + 0.784·13-s + 1.00·17-s − 1.93·19-s + 0.0725·23-s + 0.752·25-s + 0.724·29-s + 1.07·31-s + 0.245·35-s + 1.24·37-s + 0.825·41-s + 0.757·43-s + 1.56·47-s − 0.965·49-s − 0.571·53-s + 0.409·55-s + 0.631·59-s − 0.635·61-s + 1.03·65-s + 0.422·67-s − 0.752·71-s − 0.107·73-s + 0.0572·77-s − 0.803·79-s + 1.07·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.646217323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.646217323\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 74 T + p^{5} T^{2} \) |
| 7 | \( 1 - 24 T + p^{5} T^{2} \) |
| 11 | \( 1 - 124 T + p^{5} T^{2} \) |
| 13 | \( 1 - 478 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1198 T + p^{5} T^{2} \) |
| 19 | \( 1 + 3044 T + p^{5} T^{2} \) |
| 23 | \( 1 - 8 p T + p^{5} T^{2} \) |
| 29 | \( 1 - 3282 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5728 T + p^{5} T^{2} \) |
| 37 | \( 1 - 10326 T + p^{5} T^{2} \) |
| 41 | \( 1 - 8886 T + p^{5} T^{2} \) |
| 43 | \( 1 - 9188 T + p^{5} T^{2} \) |
| 47 | \( 1 - 23664 T + p^{5} T^{2} \) |
| 53 | \( 1 + 11686 T + p^{5} T^{2} \) |
| 59 | \( 1 - 16876 T + p^{5} T^{2} \) |
| 61 | \( 1 + 18482 T + p^{5} T^{2} \) |
| 67 | \( 1 - 15532 T + p^{5} T^{2} \) |
| 71 | \( 1 + 31960 T + p^{5} T^{2} \) |
| 73 | \( 1 + 4886 T + p^{5} T^{2} \) |
| 79 | \( 1 + 44560 T + p^{5} T^{2} \) |
| 83 | \( 1 - 67364 T + p^{5} T^{2} \) |
| 89 | \( 1 + 71994 T + p^{5} T^{2} \) |
| 97 | \( 1 - 48866 T + p^{5} T^{2} \) |
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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
−12.33971653580569502458980432207, −10.98389762045506110061148698151, −10.14156500667642852331460817722, −9.149645799110831032486257259160, −8.103627193165375952352363227214, −6.48087681106359819061892926478, −5.78958467217186203614475353338, −4.29734355854269458189184018739, −2.53515968865354779891918608193, −1.20128249135188900276813300598,
1.20128249135188900276813300598, 2.53515968865354779891918608193, 4.29734355854269458189184018739, 5.78958467217186203614475353338, 6.48087681106359819061892926478, 8.103627193165375952352363227214, 9.149645799110831032486257259160, 10.14156500667642852331460817722, 10.98389762045506110061148698151, 12.33971653580569502458980432207
