| L(s) = 1 | − 8·2-s + 31.0·3-s + 64·4-s − 544.·5-s − 248.·6-s − 1.68e3·7-s − 512·8-s − 1.22e3·9-s + 4.35e3·10-s + 5.62e3·11-s + 1.98e3·12-s + 5.22e3·13-s + 1.34e4·14-s − 1.68e4·15-s + 4.09e3·16-s − 1.25e4·17-s + 9.79e3·18-s + 1.83e4·19-s − 3.48e4·20-s − 5.22e4·21-s − 4.50e4·22-s + 4.02e4·23-s − 1.58e4·24-s + 2.18e5·25-s − 4.18e4·26-s − 1.05e5·27-s − 1.07e5·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.663·3-s + 0.5·4-s − 1.94·5-s − 0.469·6-s − 1.85·7-s − 0.353·8-s − 0.560·9-s + 1.37·10-s + 1.27·11-s + 0.331·12-s + 0.660·13-s + 1.31·14-s − 1.29·15-s + 0.250·16-s − 0.620·17-s + 0.396·18-s + 0.614·19-s − 0.974·20-s − 1.23·21-s − 0.901·22-s + 0.689·23-s − 0.234·24-s + 2.79·25-s − 0.466·26-s − 1.03·27-s − 0.928·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6630823058\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6630823058\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 37 | \( 1 + 5.06e4T \) |
| good | 3 | \( 1 - 31.0T + 2.18e3T^{2} \) |
| 5 | \( 1 + 544.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.68e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.62e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 5.22e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.25e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.83e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.02e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.87e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.69e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + 2.64e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.06e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.75e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.82e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.24e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.25e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.66e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.62e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.68e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.63e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 6.71e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.16e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.63e7T + 8.07e13T^{2} \) |
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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
−12.92684685138287738842199322364, −11.87467077406118727389709149179, −10.99229314924797784236458797344, −9.294355666150829675035190826528, −8.725634353394088226981647445771, −7.43043440075244597754666099101, −6.45934762301571968344495601521, −3.77905155872665571343701753993, −3.14087168157851652116598532855, −0.56121230674389349300985486201,
0.56121230674389349300985486201, 3.14087168157851652116598532855, 3.77905155872665571343701753993, 6.45934762301571968344495601521, 7.43043440075244597754666099101, 8.725634353394088226981647445771, 9.294355666150829675035190826528, 10.99229314924797784236458797344, 11.87467077406118727389709149179, 12.92684685138287738842199322364
