| L(s) = 1 | + 21.6·2-s + 79.0·3-s + 340.·4-s − 120.·5-s + 1.71e3·6-s + 240.·7-s + 4.58e3·8-s + 4.06e3·9-s − 2.61e3·10-s − 1.08e3·11-s + 2.68e4·12-s + 5.19e3·14-s − 9.54e3·15-s + 5.57e4·16-s − 2.73e4·17-s + 8.79e4·18-s + 2.50e4·19-s − 4.10e4·20-s + 1.89e4·21-s − 2.35e4·22-s − 8.02e4·23-s + 3.62e5·24-s − 6.35e4·25-s + 1.48e5·27-s + 8.16e4·28-s − 1.19e5·29-s − 2.06e5·30-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 1.69·3-s + 2.65·4-s − 0.431·5-s + 3.23·6-s + 0.264·7-s + 3.16·8-s + 1.85·9-s − 0.825·10-s − 0.246·11-s + 4.49·12-s + 0.505·14-s − 0.730·15-s + 3.40·16-s − 1.35·17-s + 3.55·18-s + 0.838·19-s − 1.14·20-s + 0.447·21-s − 0.470·22-s − 1.37·23-s + 5.35·24-s − 0.813·25-s + 1.45·27-s + 0.702·28-s − 0.906·29-s − 1.39·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(12.23568569\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.23568569\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 - 21.6T + 128T^{2} \) |
| 3 | \( 1 - 79.0T + 2.18e3T^{2} \) |
| 5 | \( 1 + 120.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 240.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.08e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 2.73e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.50e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.02e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.19e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.04e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.25e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 8.21e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.26e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.88e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.41e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.19e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.41e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.71e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.44e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.13e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.32e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.21e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.06e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.98e6T + 8.07e13T^{2} \) |
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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
−11.83738910992813461360139722124, −10.83389827157981905805865815837, −9.406152926247818840707622743788, −7.937203041030131267969671390986, −7.36915341007847522572208115392, −5.96029222504225143025388329799, −4.44156900137458239063504360155, −3.81231990917465660473066492483, −2.69952814947173114174843857814, −1.88054623949308911396458729081,
1.88054623949308911396458729081, 2.69952814947173114174843857814, 3.81231990917465660473066492483, 4.44156900137458239063504360155, 5.96029222504225143025388329799, 7.36915341007847522572208115392, 7.937203041030131267969671390986, 9.406152926247818840707622743788, 10.83389827157981905805865815837, 11.83738910992813461360139722124
