| L(s) = 1 | + 8·2-s + 27·3-s + 64·4-s − 220.·5-s + 216·6-s − 343·7-s + 512·8-s + 729·9-s − 1.76e3·10-s − 8.37e3·11-s + 1.72e3·12-s − 2.19e3·13-s − 2.74e3·14-s − 5.96e3·15-s + 4.09e3·16-s + 2.12e4·17-s + 5.83e3·18-s − 4.94e4·19-s − 1.41e4·20-s − 9.26e3·21-s − 6.70e4·22-s − 528.·23-s + 1.38e4·24-s − 2.93e4·25-s − 1.75e4·26-s + 1.96e4·27-s − 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.789·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.558·10-s − 1.89·11-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.456·15-s + 0.250·16-s + 1.04·17-s + 0.235·18-s − 1.65·19-s − 0.394·20-s − 0.218·21-s − 1.34·22-s − 0.00905·23-s + 0.204·24-s − 0.376·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.674722187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.674722187\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 3 | \( 1 - 27T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 5 | \( 1 + 220.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 8.37e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.12e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.94e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 528.T + 3.40e9T^{2} \) |
| 29 | \( 1 - 7.98e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.52e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.40e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 3.60e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.14e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.26e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.44e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 7.57e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 2.90e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 7.77e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.97e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.23e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 9.63e5T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.88e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.11e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.84e5T + 8.07e13T^{2} \) |
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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
−9.983787557984212237539221384446, −8.362712290606232130531095221974, −7.984911160143960383975623128010, −7.04506999541625635867326155950, −5.93257025119981743242469349736, −4.86227520211358183461561480101, −4.00910992507389868785749469408, −2.95222725432103391948700319052, −2.29809633194729612560148835511, −0.58359868310935756728320140334,
0.58359868310935756728320140334, 2.29809633194729612560148835511, 2.95222725432103391948700319052, 4.00910992507389868785749469408, 4.86227520211358183461561480101, 5.93257025119981743242469349736, 7.04506999541625635867326155950, 7.984911160143960383975623128010, 8.362712290606232130531095221974, 9.983787557984212237539221384446
