| L(s) = 1 | − 14.0·2-s + 27·3-s + 68.6·4-s − 378.·6-s − 1.07e3·7-s + 832.·8-s + 729·9-s + 1.67e3·11-s + 1.85e3·12-s − 552.·13-s + 1.51e4·14-s − 2.04e4·16-s − 2.70e4·17-s − 1.02e4·18-s + 3.43e4·19-s − 2.90e4·21-s − 2.34e4·22-s − 8.16e4·23-s + 2.24e4·24-s + 7.74e3·26-s + 1.96e4·27-s − 7.39e4·28-s + 6.56e4·29-s + 2.85e5·31-s + 1.80e5·32-s + 4.52e4·33-s + 3.79e5·34-s + ⋯ |
| L(s) = 1 | − 1.23·2-s + 0.577·3-s + 0.536·4-s − 0.715·6-s − 1.18·7-s + 0.574·8-s + 0.333·9-s + 0.379·11-s + 0.309·12-s − 0.0697·13-s + 1.47·14-s − 1.24·16-s − 1.33·17-s − 0.413·18-s + 1.15·19-s − 0.685·21-s − 0.470·22-s − 1.39·23-s + 0.331·24-s + 0.0864·26-s + 0.192·27-s − 0.637·28-s + 0.499·29-s + 1.72·31-s + 0.973·32-s + 0.218·33-s + 1.65·34-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
| \(L(4)\) |
\(\approx\) |
\(0.9151764615\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9151764615\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 + 14.0T + 128T^{2} \) |
| 7 | \( 1 + 1.07e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.67e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 552.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.70e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.43e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.16e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.56e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.85e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.85e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.43e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.23e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.26e6T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.30e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.25e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.60e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.27e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 4.49e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 5.52e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.64e4T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.86e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.36e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.06e7T + 8.07e13T^{2} \) |
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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
−13.24936199882757953670365475058, −11.84111341269738443003595408112, −10.34087768296664603198948003554, −9.567152135103618210766250341392, −8.751555351736226461559661597720, −7.55734553773291909537195326786, −6.39294018876178700159001807771, −4.17915684217684545680841790673, −2.48708345884470647142655602463, −0.73939656655417857463799556133,
0.73939656655417857463799556133, 2.48708345884470647142655602463, 4.17915684217684545680841790673, 6.39294018876178700159001807771, 7.55734553773291909537195326786, 8.751555351736226461559661597720, 9.567152135103618210766250341392, 10.34087768296664603198948003554, 11.84111341269738443003595408112, 13.24936199882757953670365475058
