| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 12-s + 2·13-s + 14-s − 15-s + 16-s + 6·17-s + 18-s − 8·19-s + 20-s − 21-s − 24-s + 25-s + 2·26-s − 27-s + 28-s − 30-s + 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.83·19-s + 0.223·20-s − 0.218·21-s − 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s − 0.182·30-s + 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.244020341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.244020341\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.13099109383514, −12.61152874031487, −12.44648127338623, −11.79581780451425, −11.29880702521282, −10.85418385665603, −10.56725254663535, −9.942550041004715, −9.532361935576713, −8.879052757316177, −8.274205484472127, −7.779027807063295, −7.434671536464343, −6.508655586011077, −6.260054492255831, −5.952867439261224, −5.297025755079636, −4.803287552979304, −4.302711486808509, −3.816294752769613, −3.144916390278746, −2.437711955030131, −1.957424991436991, −1.167582479999241, −0.6586203238922446,
0.6586203238922446, 1.167582479999241, 1.957424991436991, 2.437711955030131, 3.144916390278746, 3.816294752769613, 4.302711486808509, 4.803287552979304, 5.297025755079636, 5.952867439261224, 6.260054492255831, 6.508655586011077, 7.434671536464343, 7.779027807063295, 8.274205484472127, 8.879052757316177, 9.532361935576713, 9.942550041004715, 10.56725254663535, 10.85418385665603, 11.29880702521282, 11.79581780451425, 12.44648127338623, 12.61152874031487, 13.13099109383514
