L(s) = 1 | − 5.06e4·3-s + 2.37e6·5-s − 1.69e7·7-s + 1.40e9·9-s + 1.62e7·11-s − 5.04e10·13-s − 1.20e11·15-s + 2.25e11·17-s + 1.71e12·19-s + 8.56e11·21-s + 1.40e13·23-s − 1.34e13·25-s − 1.22e13·27-s − 1.13e12·29-s − 1.04e14·31-s − 8.21e11·33-s − 4.02e13·35-s + 1.69e14·37-s + 2.55e15·39-s − 3.30e15·41-s − 1.12e15·43-s + 3.33e15·45-s + 3.49e15·47-s − 1.11e16·49-s − 1.14e16·51-s − 2.99e16·53-s + 3.85e13·55-s + ⋯ |
L(s) = 1 | − 1.48·3-s + 0.544·5-s − 0.158·7-s + 1.20·9-s + 0.00207·11-s − 1.31·13-s − 0.808·15-s + 0.460·17-s + 1.21·19-s + 0.235·21-s + 1.62·23-s − 0.703·25-s − 0.308·27-s − 0.0145·29-s − 0.710·31-s − 0.00308·33-s − 0.0862·35-s + 0.214·37-s + 1.95·39-s − 1.57·41-s − 0.342·43-s + 0.657·45-s + 0.456·47-s − 0.974·49-s − 0.683·51-s − 1.24·53-s + 0.00112·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(10)\) |
\(\approx\) |
\(0.9769289664\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9769289664\) |
\(L(\frac{21}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + 1876 p^{3} T + p^{19} T^{2} \) |
| 5 | \( 1 - 475482 p T + p^{19} T^{2} \) |
| 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 | \( 1 + 635013222218448094 T + p^{19} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.30139658906966558257206209594, −10.18538137082687661977453391286, −9.387688526107799403590231283361, −7.52510229546758450926298979142, −6.56255033234105818320739888536, −5.42479685274641551901673366227, −4.88500668343664330359261361719, −3.12296501732104396824758888180, −1.60707189225044031553948379450, −0.48858854388928965352795133584,
0.48858854388928965352795133584, 1.60707189225044031553948379450, 3.12296501732104396824758888180, 4.88500668343664330359261361719, 5.42479685274641551901673366227, 6.56255033234105818320739888536, 7.52510229546758450926298979142, 9.387688526107799403590231283361, 10.18538137082687661977453391286, 11.30139658906966558257206209594