| L(s) = 1 | − 8·2-s − 24.3·3-s + 64·4-s − 116.·5-s + 194.·6-s + 1.26e3·7-s − 512·8-s − 1.59e3·9-s + 933.·10-s + 3.68e3·11-s − 1.55e3·12-s − 1.45e4·13-s − 1.01e4·14-s + 2.84e3·15-s + 4.09e3·16-s − 2.39e4·17-s + 1.27e4·18-s − 1.48e3·19-s − 7.46e3·20-s − 3.08e4·21-s − 2.94e4·22-s + 3.46e4·23-s + 1.24e4·24-s − 6.45e4·25-s + 1.16e5·26-s + 9.21e4·27-s + 8.10e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.520·3-s + 0.5·4-s − 0.417·5-s + 0.368·6-s + 1.39·7-s − 0.353·8-s − 0.728·9-s + 0.295·10-s + 0.834·11-s − 0.260·12-s − 1.84·13-s − 0.987·14-s + 0.217·15-s + 0.250·16-s − 1.18·17-s + 0.515·18-s − 0.0495·19-s − 0.208·20-s − 0.727·21-s − 0.590·22-s + 0.593·23-s + 0.184·24-s − 0.825·25-s + 1.30·26-s + 0.900·27-s + 0.698·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.6565593186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6565593186\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 269 | \( 1 + 1.94e7T \) |
| good | 3 | \( 1 + 24.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + 116.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.26e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.68e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.45e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.39e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.48e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.46e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 4.79e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.58e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.73e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.29e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.36e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.23e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.06e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.35e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.98e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.21e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.83e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.28e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.12e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.90e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.03e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.17e7T + 8.07e13T^{2} \) |
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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
−9.493238120886446332516831831805, −8.888306376044114532218024080843, −7.80920513926631521106519575728, −7.28666583864275565658552493812, −6.10483342011451388547862282900, −5.05707603792514387455766598271, −4.24398143543010063135230630815, −2.61318125640387370724291688577, −1.68147615998146276550705001117, −0.39645870772114069840500024429,
0.39645870772114069840500024429, 1.68147615998146276550705001117, 2.61318125640387370724291688577, 4.24398143543010063135230630815, 5.05707603792514387455766598271, 6.10483342011451388547862282900, 7.28666583864275565658552493812, 7.80920513926631521106519575728, 8.888306376044114532218024080843, 9.493238120886446332516831831805
