| L(s) = 1 | − 18.7·2-s + 76.5·3-s + 222.·4-s − 1.43e3·6-s + 343·7-s − 1.76e3·8-s + 3.66e3·9-s − 5.67e3·11-s + 1.70e4·12-s − 8.26e3·13-s − 6.42e3·14-s + 4.64e3·16-s − 4.75e3·17-s − 6.86e4·18-s + 1.46e4·19-s + 2.62e4·21-s + 1.06e5·22-s + 7.79e4·23-s − 1.35e5·24-s + 1.54e5·26-s + 1.13e5·27-s + 7.63e4·28-s + 1.53e5·29-s + 2.84e5·31-s + 1.39e5·32-s − 4.34e5·33-s + 8.89e4·34-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.63·3-s + 1.73·4-s − 2.70·6-s + 0.377·7-s − 1.22·8-s + 1.67·9-s − 1.28·11-s + 2.84·12-s − 1.04·13-s − 0.625·14-s + 0.283·16-s − 0.234·17-s − 2.77·18-s + 0.490·19-s + 0.618·21-s + 2.12·22-s + 1.33·23-s − 1.99·24-s + 1.72·26-s + 1.10·27-s + 0.657·28-s + 1.16·29-s + 1.71·31-s + 0.752·32-s − 2.10·33-s + 0.388·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.656832655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.656832655\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - 343T \) |
| good | 2 | \( 1 + 18.7T + 128T^{2} \) |
| 3 | \( 1 - 76.5T + 2.18e3T^{2} \) |
| 11 | \( 1 + 5.67e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.26e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 4.75e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.46e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.79e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.53e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.84e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.73e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 8.19e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.70e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.05e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.33e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.49e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.15e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.41e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.23e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.31e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.66e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.08e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.05e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.28e7T + 8.07e13T^{2} \) |
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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
−10.85110757442881157958581170303, −10.03826639407544021096552540734, −9.245862674564724125384399411879, −8.387334335313768627453712764092, −7.76414682907167813215447482022, −6.99750772076142556273918497901, −4.81456510711975334268249514710, −2.88025542876342794767655799607, −2.25048603492425095283920901743, −0.836757993593666188594945380123,
0.836757993593666188594945380123, 2.25048603492425095283920901743, 2.88025542876342794767655799607, 4.81456510711975334268249514710, 6.99750772076142556273918497901, 7.76414682907167813215447482022, 8.387334335313768627453712764092, 9.245862674564724125384399411879, 10.03826639407544021096552540734, 10.85110757442881157958581170303
