| L(s) = 1 | − 92.4·5-s − 1.12e3·7-s − 1.11e3·11-s − 9.93e3·13-s + 3.15e4·17-s − 1.90e4·19-s − 4.77e4·23-s − 6.95e4·25-s − 2.52e5·29-s − 4.71e4·31-s + 1.03e5·35-s − 6.27e4·37-s + 5.38e5·41-s − 1.90e5·43-s − 2.91e5·47-s + 4.33e5·49-s + 2.17e5·53-s + 1.03e5·55-s − 2.01e6·59-s − 1.05e6·61-s + 9.18e5·65-s + 1.46e6·67-s − 1.77e6·71-s + 5.31e6·73-s + 1.24e6·77-s + 1.15e6·79-s − 1.53e6·83-s + ⋯ |
| L(s) = 1 | − 0.330·5-s − 1.23·7-s − 0.252·11-s − 1.25·13-s + 1.55·17-s − 0.636·19-s − 0.818·23-s − 0.890·25-s − 1.92·29-s − 0.283·31-s + 0.408·35-s − 0.203·37-s + 1.21·41-s − 0.365·43-s − 0.410·47-s + 0.526·49-s + 0.200·53-s + 0.0834·55-s − 1.27·59-s − 0.593·61-s + 0.414·65-s + 0.595·67-s − 0.588·71-s + 1.59·73-s + 0.311·77-s + 0.262·79-s − 0.295·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6113367747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6113367747\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 92.4T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.12e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 1.11e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 9.93e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.15e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.90e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.77e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 2.52e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 4.71e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 6.27e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 5.38e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.90e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.91e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.17e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.01e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.05e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.46e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.77e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.31e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.15e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.53e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.43e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.94e6T + 8.07e13T^{2} \) |
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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
−9.794579984979057925721949698082, −9.407645518156349050130896497179, −7.933999233113829538406417368624, −7.37191168574205809563219653951, −6.20314184171773519077929062261, −5.36218189603456539898778022066, −4.01089532889130371579139370678, −3.16085054699602505331867057873, −1.99275734558051534311559880620, −0.33414357540699778441641663664,
0.33414357540699778441641663664, 1.99275734558051534311559880620, 3.16085054699602505331867057873, 4.01089532889130371579139370678, 5.36218189603456539898778022066, 6.20314184171773519077929062261, 7.37191168574205809563219653951, 7.933999233113829538406417368624, 9.407645518156349050130896497179, 9.794579984979057925721949698082
