| L(s) = 1 | − 3.66·2-s + 42.3·3-s − 114.·4-s − 251.·5-s − 155.·6-s − 343·7-s + 888.·8-s − 393.·9-s + 921.·10-s − 2.92e3·11-s − 4.85e3·12-s − 2.19e3·13-s + 1.25e3·14-s − 1.06e4·15-s + 1.14e4·16-s + 2.77e4·17-s + 1.44e3·18-s + 3.43e4·19-s + 2.88e4·20-s − 1.45e4·21-s + 1.07e4·22-s − 2.16e4·23-s + 3.76e4·24-s − 1.47e4·25-s + 8.04e3·26-s − 1.09e5·27-s + 3.93e4·28-s + ⋯ |
| L(s) = 1 | − 0.323·2-s + 0.905·3-s − 0.895·4-s − 0.900·5-s − 0.293·6-s − 0.377·7-s + 0.613·8-s − 0.180·9-s + 0.291·10-s − 0.662·11-s − 0.810·12-s − 0.277·13-s + 0.122·14-s − 0.815·15-s + 0.696·16-s + 1.37·17-s + 0.0583·18-s + 1.14·19-s + 0.806·20-s − 0.342·21-s + 0.214·22-s − 0.371·23-s + 0.555·24-s − 0.188·25-s + 0.0897·26-s − 1.06·27-s + 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.184050826\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.184050826\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 2 | \( 1 + 3.66T + 128T^{2} \) |
| 3 | \( 1 - 42.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + 251.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 2.92e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.77e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.43e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.16e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.15e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.30e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.81e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.16e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 7.51e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.00e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 4.66e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.34e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.22e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.36e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 6.19e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.50e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.88e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 5.10e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.60e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 8.11e5T + 8.07e13T^{2} \) |
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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
−12.79855601663954017464238655390, −11.69301434076027506584294673142, −10.12667583731171861481096510045, −9.308793133193854476623161794553, −8.070774301789460284001701249687, −7.66859782573741137923503826714, −5.47437468325190947382748799713, −3.97886541521018069206200973171, −2.90862220437325154832540882806, −0.69464762461064572171608488659,
0.69464762461064572171608488659, 2.90862220437325154832540882806, 3.97886541521018069206200973171, 5.47437468325190947382748799713, 7.66859782573741137923503826714, 8.070774301789460284001701249687, 9.308793133193854476623161794553, 10.12667583731171861481096510045, 11.69301434076027506584294673142, 12.79855601663954017464238655390
