L(s) = 1 | − 13.7·2-s − 81·3-s − 324.·4-s − 1.28e3·5-s + 1.11e3·6-s + 3.21e3·7-s + 1.14e4·8-s + 6.56e3·9-s + 1.76e4·10-s − 5.83e4·11-s + 2.62e4·12-s + 1.42e5·13-s − 4.41e4·14-s + 1.04e5·15-s + 8.86e3·16-s − 6.34e5·17-s − 8.99e4·18-s + 1.07e6·19-s + 4.17e5·20-s − 2.60e5·21-s + 7.99e5·22-s − 3.25e5·23-s − 9.28e5·24-s − 2.94e5·25-s − 1.95e6·26-s − 5.31e5·27-s − 1.04e6·28-s + ⋯ |
L(s) = 1 | − 0.605·2-s − 0.577·3-s − 0.633·4-s − 0.921·5-s + 0.349·6-s + 0.506·7-s + 0.989·8-s + 0.333·9-s + 0.558·10-s − 1.20·11-s + 0.365·12-s + 1.38·13-s − 0.306·14-s + 0.532·15-s + 0.0338·16-s − 1.84·17-s − 0.201·18-s + 1.88·19-s + 0.583·20-s − 0.292·21-s + 0.727·22-s − 0.242·23-s − 0.571·24-s − 0.150·25-s − 0.839·26-s − 0.192·27-s − 0.320·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.3899076537\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3899076537\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 - 1.21e7T \) |
good | 2 | \( 1 + 13.7T + 512T^{2} \) |
| 5 | \( 1 + 1.28e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 3.21e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.83e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.42e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 6.34e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.07e6T + 3.22e11T^{2} \) |
| 23 | \( 1 + 3.25e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.66e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.51e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.62e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.02e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.84e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.59e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.97e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 5.83e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.77e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.90e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.70e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.31e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.84e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.00e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.11e8T + 7.60e17T^{2} \) |
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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
−11.16123893431053354140192342177, −10.05840432238839689240206349657, −8.870844765286802037011236111921, −8.029922582840290404141364324692, −7.23707499208858417159593949925, −5.60300641742134225401241013131, −4.63237004438984067434644815502, −3.59114610158747880728115843431, −1.62356764506891765591320638164, −0.36053967947958405281710988710,
0.36053967947958405281710988710, 1.62356764506891765591320638164, 3.59114610158747880728115843431, 4.63237004438984067434644815502, 5.60300641742134225401241013131, 7.23707499208858417159593949925, 8.029922582840290404141364324692, 8.870844765286802037011236111921, 10.05840432238839689240206349657, 11.16123893431053354140192342177