| L(s) = 1 | − 456·2-s − 5.06e4·3-s − 3.16e5·4-s + 2.30e7·6-s + 1.69e7·7-s + 3.83e8·8-s + 1.40e9·9-s − 1.62e7·11-s + 1.60e10·12-s − 5.04e10·13-s − 7.71e9·14-s − 8.93e9·16-s − 2.25e11·17-s − 6.39e11·18-s − 1.71e12·19-s − 8.56e11·21-s + 7.39e9·22-s − 1.40e13·23-s − 1.94e13·24-s + 2.29e13·26-s − 1.22e13·27-s − 5.35e12·28-s + 1.13e12·29-s − 1.04e14·31-s − 1.96e14·32-s + 8.21e11·33-s + 1.02e14·34-s + ⋯ |
| L(s) = 1 | − 0.629·2-s − 1.48·3-s − 0.603·4-s + 0.935·6-s + 0.158·7-s + 1.00·8-s + 1.20·9-s − 0.00207·11-s + 0.896·12-s − 1.31·13-s − 0.0997·14-s − 0.0325·16-s − 0.460·17-s − 0.760·18-s − 1.21·19-s − 0.235·21-s + 0.00130·22-s − 1.62·23-s − 1.50·24-s + 0.830·26-s − 0.308·27-s − 0.0956·28-s + 0.0145·29-s − 0.710·31-s − 0.989·32-s + 0.00308·33-s + 0.289·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(0.05880789425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05880789425\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 57 p^{3} T + p^{19} T^{2} \) |
| 3 | \( 1 + 1876 p^{3} T + p^{19} T^{2} \) |
| 7 | \( 1 - 345256 p^{2} T + p^{19} T^{2} \) |
| 11 | \( 1 + 1473828 p T + p^{19} T^{2} \) |
| 13 | \( 1 + 3878585774 p T + p^{19} T^{2} \) |
| 17 | \( 1 + 13239417618 p T + p^{19} T^{2} \) |
| 19 | \( 1 + 1710278572660 T + p^{19} T^{2} \) |
| 23 | \( 1 + 14036534788872 T + p^{19} T^{2} \) |
| 29 | \( 1 - 1137835269510 T + p^{19} T^{2} \) |
| 31 | \( 1 + 104626880141728 T + p^{19} T^{2} \) |
| 37 | \( 1 - 169392327370594 T + p^{19} T^{2} \) |
| 41 | \( 1 + 3309984750560838 T + p^{19} T^{2} \) |
| 43 | \( 1 + 1127913532193492 T + p^{19} T^{2} \) |
| 47 | \( 1 + 3498693987674256 T + p^{19} T^{2} \) |
| 53 | \( 1 + 29956294112980302 T + p^{19} T^{2} \) |
| 59 | \( 1 - 58391397642732420 T + p^{19} T^{2} \) |
| 61 | \( 1 - 23373685132672742 T + p^{19} T^{2} \) |
| 67 | \( 1 - 205102524257382244 T + p^{19} T^{2} \) |
| 71 | \( 1 + 177902341950417768 T + p^{19} T^{2} \) |
| 73 | \( 1 + 299853775038660122 T + p^{19} T^{2} \) |
| 79 | \( 1 + 92227090144007440 T + p^{19} T^{2} \) |
| 83 | \( 1 + 1208542823470585932 T + p^{19} T^{2} \) |
| 89 | \( 1 - 4371201192290304330 T + p^{19} T^{2} \) |
| 97 | \( 1 - 635013222218448094 T + p^{19} T^{2} \) |
| 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
−12.98807865864380456973285840405, −11.89333192726979383462894830115, −10.64864378166973429429096728963, −9.731585230808419504976260343588, −8.152974869652894877409251675614, −6.69105835493309909005498239720, −5.27023166570401725154137539486, −4.31327029616242747788395315478, −1.78286092977548508436638351642, −0.15550265087099468488939272061,
0.15550265087099468488939272061, 1.78286092977548508436638351642, 4.31327029616242747788395315478, 5.27023166570401725154137539486, 6.69105835493309909005498239720, 8.152974869652894877409251675614, 9.731585230808419504976260343588, 10.64864378166973429429096728963, 11.89333192726979383462894830115, 12.98807865864380456973285840405
