| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 2·7-s − 8-s + 9-s − 12-s − 4·13-s − 2·14-s + 16-s − 18-s + 8·19-s − 2·21-s − 6·23-s + 24-s − 5·25-s + 4·26-s − 27-s + 2·28-s − 2·29-s − 4·31-s − 32-s + 36-s + 6·37-s − 8·38-s + 4·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.235·18-s + 1.83·19-s − 0.436·21-s − 1.25·23-s + 0.204·24-s − 25-s + 0.784·26-s − 0.192·27-s + 0.377·28-s − 0.371·29-s − 0.718·31-s − 0.176·32-s + 1/6·36-s + 0.986·37-s − 1.29·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209814 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.489094195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.489094195\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| 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
−12.99416839489665, −12.30159075341659, −11.92562819662867, −11.58254648743666, −11.22088942171208, −10.73826879792061, −10.01648931395514, −9.733460535454705, −9.508317562884972, −8.788233267483093, −8.142490188961175, −7.740997256849046, −7.330101386028893, −7.102645108328672, −6.127272527638084, −5.847139027387512, −5.267787431714234, −4.911424938835571, −4.059544920858206, −3.780618116501469, −2.816612440236282, −2.261946096234839, −1.786357998108844, −0.9836243918129483, −0.4689786043456201,
0.4689786043456201, 0.9836243918129483, 1.786357998108844, 2.261946096234839, 2.816612440236282, 3.780618116501469, 4.059544920858206, 4.911424938835571, 5.267787431714234, 5.847139027387512, 6.127272527638084, 7.102645108328672, 7.330101386028893, 7.740997256849046, 8.142490188961175, 8.788233267483093, 9.508317562884972, 9.733460535454705, 10.01648931395514, 10.73826879792061, 11.22088942171208, 11.58254648743666, 11.92562819662867, 12.30159075341659, 12.99416839489665
