| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 4·7-s − 3·8-s + 9-s + 12-s − 6·13-s + 4·14-s − 16-s − 2·17-s + 18-s + 8·19-s − 4·21-s + 4·23-s + 3·24-s − 6·26-s − 27-s − 4·28-s + 29-s + 4·31-s + 5·32-s − 2·34-s − 36-s − 6·37-s + 8·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.288·12-s − 1.66·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.83·19-s − 0.872·21-s + 0.834·23-s + 0.612·24-s − 1.17·26-s − 0.192·27-s − 0.755·28-s + 0.185·29-s + 0.718·31-s + 0.883·32-s − 0.342·34-s − 1/6·36-s − 0.986·37-s + 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.870044342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.870044342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−9.222941065676694823607553312826, −8.135162369876425422842333574493, −7.52809492722105176267465092732, −6.64636306414511198799043675308, −5.46976457088803416030066802406, −4.93727986250653983970388710162, −4.67243159268150509585054413330, −3.43077360977130429249437376512, −2.27666423911977039277785499969, −0.850594550508520107148863235659,
0.850594550508520107148863235659, 2.27666423911977039277785499969, 3.43077360977130429249437376512, 4.67243159268150509585054413330, 4.93727986250653983970388710162, 5.46976457088803416030066802406, 6.64636306414511198799043675308, 7.52809492722105176267465092732, 8.135162369876425422842333574493, 9.222941065676694823607553312826
