| L(s) = 1 | − 17.4·2-s − 81·3-s − 206.·4-s + 625·5-s + 1.41e3·6-s − 3.26e3·7-s + 1.25e4·8-s + 6.56e3·9-s − 1.09e4·10-s − 1.63e4·11-s + 1.67e4·12-s − 1.68e5·13-s + 5.71e4·14-s − 5.06e4·15-s − 1.13e5·16-s − 3.97e5·17-s − 1.14e5·18-s − 1.30e5·19-s − 1.28e5·20-s + 2.64e5·21-s + 2.85e5·22-s + 1.77e6·23-s − 1.01e6·24-s + 3.90e5·25-s + 2.95e6·26-s − 5.31e5·27-s + 6.74e5·28-s + ⋯ |
| L(s) = 1 | − 0.772·2-s − 0.577·3-s − 0.403·4-s + 0.447·5-s + 0.446·6-s − 0.514·7-s + 1.08·8-s + 0.333·9-s − 0.345·10-s − 0.335·11-s + 0.232·12-s − 1.63·13-s + 0.397·14-s − 0.258·15-s − 0.434·16-s − 1.15·17-s − 0.257·18-s − 0.229·19-s − 0.180·20-s + 0.297·21-s + 0.259·22-s + 1.32·23-s − 0.625·24-s + 0.200·25-s + 1.26·26-s − 0.192·27-s + 0.207·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.2617874723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2617874723\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 2 | \( 1 + 17.4T + 512T^{2} \) |
| 7 | \( 1 + 3.26e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.63e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.68e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.97e5T + 1.18e11T^{2} \) |
| 23 | \( 1 - 1.77e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.10e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.49e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.40e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 7.51e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.50e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.98e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.86e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.18e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.76e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 7.82e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.99e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.70e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 5.51e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 2.14e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.76e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.58e8T + 7.60e17T^{2} \) |
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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
−10.14470189346395547465108766987, −9.411199191645607592367966533217, −8.562996871454934725499972110786, −7.31916188568729187321102473693, −6.57788422435322637249609859724, −5.13972122795359851457945703855, −4.56112284399463518974203783124, −2.84268216709228098473908879856, −1.58677567998265041905642075103, −0.26770276494045140076290568463,
0.26770276494045140076290568463, 1.58677567998265041905642075103, 2.84268216709228098473908879856, 4.56112284399463518974203783124, 5.13972122795359851457945703855, 6.57788422435322637249609859724, 7.31916188568729187321102473693, 8.562996871454934725499972110786, 9.411199191645607592367966533217, 10.14470189346395547465108766987
