L(s) = 1 | − 43.9·2-s + 81·3-s + 1.41e3·4-s − 993.·5-s − 3.55e3·6-s + 3.42e3·7-s − 3.97e4·8-s + 6.56e3·9-s + 4.36e4·10-s − 7.50e4·11-s + 1.14e5·12-s + 6.68e4·13-s − 1.50e5·14-s − 8.04e4·15-s + 1.01e6·16-s + 1.99e5·17-s − 2.88e5·18-s + 7.98e4·19-s − 1.40e6·20-s + 2.77e5·21-s + 3.29e6·22-s + 2.06e6·23-s − 3.21e6·24-s − 9.65e5·25-s − 2.93e6·26-s + 5.31e5·27-s + 4.85e6·28-s + ⋯ |
L(s) = 1 | − 1.94·2-s + 0.577·3-s + 2.76·4-s − 0.710·5-s − 1.12·6-s + 0.539·7-s − 3.42·8-s + 0.333·9-s + 1.37·10-s − 1.54·11-s + 1.59·12-s + 0.649·13-s − 1.04·14-s − 0.410·15-s + 3.88·16-s + 0.579·17-s − 0.646·18-s + 0.140·19-s − 1.96·20-s + 0.311·21-s + 3.00·22-s + 1.53·23-s − 1.97·24-s − 0.494·25-s − 1.26·26-s + 0.192·27-s + 1.49·28-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
\(L(5)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{11}{2})\) |
|
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 | \( 1 + 43.9T + 512T^{2} \) |
| 5 | \( 1 + 993.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 3.42e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 7.50e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 6.68e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.99e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.98e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.06e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 5.68e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 1.44e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 6.74e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.10e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.99e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.76e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 9.14e7T + 3.29e15T^{2} \) |
| 61 | \( 1 + 3.45e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 4.20e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.88e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.61e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.32e6T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.97e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.53e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.40e8T + 7.60e17T^{2} \) |
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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
−10.44755886772608185990956372730, −9.315003990342768542457619820726, −8.409581232044859424212407731590, −7.78050817225953369528589513410, −7.12180089745040088697708533172, −5.48532259908728932985461746995, −3.40960867608328551621351779029, −2.31978706073555138836777960943, −1.14246781129545837827254691319, 0,
1.14246781129545837827254691319, 2.31978706073555138836777960943, 3.40960867608328551621351779029, 5.48532259908728932985461746995, 7.12180089745040088697708533172, 7.78050817225953369528589513410, 8.409581232044859424212407731590, 9.315003990342768542457619820726, 10.44755886772608185990956372730