L(s) = 1 | + 8·2-s − 27·3-s + 64·4-s − 125·5-s − 216·6-s + 1.66e3·7-s + 512·8-s + 729·9-s − 1.00e3·10-s + 4.83e3·11-s − 1.72e3·12-s + 2.19e3·13-s + 1.33e4·14-s + 3.37e3·15-s + 4.09e3·16-s + 2.40e4·17-s + 5.83e3·18-s − 2.64e3·19-s − 8.00e3·20-s − 4.50e4·21-s + 3.86e4·22-s + 5.16e4·23-s − 1.38e4·24-s + 1.56e4·25-s + 1.75e4·26-s − 1.96e4·27-s + 1.06e5·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.83·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.09·11-s − 0.288·12-s + 0.277·13-s + 1.30·14-s + 0.258·15-s + 0.250·16-s + 1.18·17-s + 0.235·18-s − 0.0885·19-s − 0.223·20-s − 1.06·21-s + 0.773·22-s + 0.885·23-s − 0.204·24-s + 0.199·25-s + 0.196·26-s − 0.192·27-s + 0.919·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 390 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(4.293640383\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.293640383\) |
\(L(\frac{9}{2})\) |
|
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 | \( 1 + 27T \) |
| 5 | \( 1 + 125T \) |
| 13 | \( 1 - 2.19e3T \) |
good | 7 | \( 1 - 1.66e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 4.83e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 2.40e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.64e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.16e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.16e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.01e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.32e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.19e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.17e4T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.07e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.53e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.93e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 6.42e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.69e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.93e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.51e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.98e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.27e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.02e7T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.25e7T + 8.07e13T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.58847445102341064631292432571, −9.158139501074715531497067789199, −8.032993444756089780505737302646, −7.32077777092540648422273746004, −6.15827947639402046688865517048, −5.14186076081049694371345070585, −4.46434901282969791711815766393, −3.43245517147617162295381955746, −1.73855107083275179480818088591, −0.991963731885094117360851227286,
0.991963731885094117360851227286, 1.73855107083275179480818088591, 3.43245517147617162295381955746, 4.46434901282969791711815766393, 5.14186076081049694371345070585, 6.15827947639402046688865517048, 7.32077777092540648422273746004, 8.032993444756089780505737302646, 9.158139501074715531497067789199, 10.58847445102341064631292432571