| L(s) = 1 | − 17.7·2-s + 27·3-s + 188.·4-s − 480.·6-s + 1.23e3·7-s − 1.06e3·8-s + 729·9-s + 1.21e3·11-s + 5.08e3·12-s + 1.35e4·13-s − 2.19e4·14-s − 5.06e3·16-s − 1.20e4·17-s − 1.29e4·18-s − 3.28e4·19-s + 3.32e4·21-s − 2.16e4·22-s − 9.42e3·23-s − 2.88e4·24-s − 2.40e5·26-s + 1.96e4·27-s + 2.32e5·28-s − 7.99e4·29-s + 1.13e5·31-s + 2.26e5·32-s + 3.28e4·33-s + 2.15e5·34-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 0.577·3-s + 1.47·4-s − 0.907·6-s + 1.35·7-s − 0.738·8-s + 0.333·9-s + 0.275·11-s + 0.848·12-s + 1.70·13-s − 2.13·14-s − 0.308·16-s − 0.597·17-s − 0.523·18-s − 1.09·19-s + 0.784·21-s − 0.432·22-s − 0.161·23-s − 0.426·24-s − 2.68·26-s + 0.192·27-s + 1.99·28-s − 0.608·29-s + 0.684·31-s + 1.22·32-s + 0.158·33-s + 0.938·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.377128648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.377128648\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 17.7T + 128T^{2} \) |
| 7 | \( 1 - 1.23e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.21e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.35e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.20e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.28e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 9.42e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 7.99e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.13e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.18e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.04e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.91e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.09e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.05e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.44e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.00e4T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.79e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.66e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.67e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.76e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.77e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.41e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.60e6T + 8.07e13T^{2} \) |
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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
−13.16048583792136995473063082095, −11.31305711574251692394688077332, −10.85588843840413736494873075116, −9.408998257847898540548208918905, −8.446569554205313574028316723086, −7.924635135958612749838947482332, −6.41225730896992552558112744786, −4.26288130300957430163256384767, −2.10417645883994214529100837019, −1.03805817675042484863550453259,
1.03805817675042484863550453259, 2.10417645883994214529100837019, 4.26288130300957430163256384767, 6.41225730896992552558112744786, 7.924635135958612749838947482332, 8.446569554205313574028316723086, 9.408998257847898540548208918905, 10.85588843840413736494873075116, 11.31305711574251692394688077332, 13.16048583792136995473063082095
