| L(s) = 1 | − 6·3-s − 25·5-s + 244·7-s − 207·9-s − 794·11-s − 169·13-s + 150·15-s − 1.53e3·17-s − 2.70e3·19-s − 1.46e3·21-s + 702·23-s + 625·25-s + 2.70e3·27-s − 5.03e3·29-s + 3.63e3·31-s + 4.76e3·33-s − 6.10e3·35-s − 7.05e3·37-s + 1.01e3·39-s − 294·41-s − 7.61e3·43-s + 5.17e3·45-s + 3.02e3·47-s + 4.27e4·49-s + 9.20e3·51-s + 626·53-s + 1.98e4·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s − 0.447·5-s + 1.88·7-s − 0.851·9-s − 1.97·11-s − 0.277·13-s + 0.172·15-s − 1.28·17-s − 1.71·19-s − 0.724·21-s + 0.276·23-s + 1/5·25-s + 0.712·27-s − 1.11·29-s + 0.679·31-s + 0.761·33-s − 0.841·35-s − 0.847·37-s + 0.106·39-s − 0.0273·41-s − 0.628·43-s + 0.380·45-s + 0.199·47-s + 2.54·49-s + 0.495·51-s + 0.0306·53-s + 0.884·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6866668734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6866668734\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| 13 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + 2 p T + p^{5} T^{2} \) |
| 7 | \( 1 - 244 T + p^{5} T^{2} \) |
| 11 | \( 1 + 794 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1534 T + p^{5} T^{2} \) |
| 19 | \( 1 + 2706 T + p^{5} T^{2} \) |
| 23 | \( 1 - 702 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5038 T + p^{5} T^{2} \) |
| 31 | \( 1 - 3634 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7058 T + p^{5} T^{2} \) |
| 41 | \( 1 + 294 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7618 T + p^{5} T^{2} \) |
| 47 | \( 1 - 3020 T + p^{5} T^{2} \) |
| 53 | \( 1 - 626 T + p^{5} T^{2} \) |
| 59 | \( 1 - 30066 T + p^{5} T^{2} \) |
| 61 | \( 1 + 5806 T + p^{5} T^{2} \) |
| 67 | \( 1 - 12436 T + p^{5} T^{2} \) |
| 71 | \( 1 + 4734 T + p^{5} T^{2} \) |
| 73 | \( 1 + 14694 T + p^{5} T^{2} \) |
| 79 | \( 1 - 39804 T + p^{5} T^{2} \) |
| 83 | \( 1 - 41776 T + p^{5} T^{2} \) |
| 89 | \( 1 - 7970 T + p^{5} T^{2} \) |
| 97 | \( 1 + 78050 T + p^{5} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.834945395764177137171426962320, −8.290895801952060228896972838795, −7.77570929332915306402699522417, −6.72392575060461851190698536394, −5.46677290140701216010770929814, −4.99000218643274992665085454103, −4.20216526791850258268763601773, −2.61082377162410966326623638176, −1.95589493830920834519066076599, −0.34454725067831903876329595703,
0.34454725067831903876329595703, 1.95589493830920834519066076599, 2.61082377162410966326623638176, 4.20216526791850258268763601773, 4.99000218643274992665085454103, 5.46677290140701216010770929814, 6.72392575060461851190698536394, 7.77570929332915306402699522417, 8.290895801952060228896972838795, 8.834945395764177137171426962320
