| L(s) = 1 | − 14.3·2-s − 27·3-s + 76.6·4-s + 348.·5-s + 386.·6-s − 1.31e3·7-s + 734.·8-s + 729·9-s − 4.99e3·10-s + 4.99e3·11-s − 2.06e3·12-s + 3.39e3·13-s + 1.88e4·14-s − 9.41e3·15-s − 2.03e4·16-s + 1.30e4·17-s − 1.04e4·18-s + 8.79e3·19-s + 2.67e4·20-s + 3.55e4·21-s − 7.14e4·22-s + 1.07e5·23-s − 1.98e4·24-s + 4.35e4·25-s − 4.86e4·26-s − 1.96e4·27-s − 1.00e5·28-s + ⋯ |
| L(s) = 1 | − 1.26·2-s − 0.577·3-s + 0.598·4-s + 1.24·5-s + 0.729·6-s − 1.45·7-s + 0.507·8-s + 0.333·9-s − 1.57·10-s + 1.13·11-s − 0.345·12-s + 0.428·13-s + 1.83·14-s − 0.720·15-s − 1.24·16-s + 0.645·17-s − 0.421·18-s + 0.294·19-s + 0.747·20-s + 0.837·21-s − 1.43·22-s + 1.84·23-s − 0.293·24-s + 0.558·25-s − 0.542·26-s − 0.192·27-s − 0.868·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.9510276257\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9510276257\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 27T \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 + 14.3T + 128T^{2} \) |
| 5 | \( 1 - 348.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.31e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.99e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 3.39e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.30e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 8.79e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 1.07e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 3.19e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.54e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 5.28e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 4.80e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.38e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.23e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.18e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 8.73e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 8.32e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.15e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.69e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.93e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.14e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.70e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.03e7T + 8.07e13T^{2} \) |
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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
−10.96092194504438051138010976746, −10.09756489850938137502089121669, −9.434388340968397339589141910600, −8.844370439997100279816956557061, −7.06090833758286326314568147375, −6.45371587881169298184934145803, −5.28123827223469590747603875338, −3.42994767035253593077028118899, −1.68464473817070593883085396031, −0.70334622483954518117393701212,
0.70334622483954518117393701212, 1.68464473817070593883085396031, 3.42994767035253593077028118899, 5.28123827223469590747603875338, 6.45371587881169298184934145803, 7.06090833758286326314568147375, 8.844370439997100279816956557061, 9.434388340968397339589141910600, 10.09756489850938137502089121669, 10.96092194504438051138010976746
