L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s + 216·6-s − 1.60e3·7-s − 512·8-s + 729·9-s − 2.20e3·11-s − 1.72e3·12-s − 5.73e3·13-s + 1.28e4·14-s + 4.09e3·16-s − 1.56e4·17-s − 5.83e3·18-s − 1.96e4·19-s + 4.33e4·21-s + 1.76e4·22-s + 2.85e4·23-s + 1.38e4·24-s + 4.59e4·26-s − 1.96e4·27-s − 1.02e5·28-s − 1.40e5·29-s − 2.91e5·31-s − 3.27e4·32-s + 5.96e4·33-s + 1.25e5·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.76·7-s − 0.353·8-s + 1/3·9-s − 0.500·11-s − 0.288·12-s − 0.724·13-s + 1.24·14-s + 1/4·16-s − 0.772·17-s − 0.235·18-s − 0.657·19-s + 1.02·21-s + 0.353·22-s + 0.488·23-s + 0.204·24-s + 0.512·26-s − 0.192·27-s − 0.883·28-s − 1.06·29-s − 1.75·31-s − 0.176·32-s + 0.288·33-s + 0.546·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.2304133662\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2304133662\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p^{3} T \) |
| 3 | \( 1 + p^{3} T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 1604 T + p^{7} T^{2} \) |
| 11 | \( 1 + 2208 T + p^{7} T^{2} \) |
| 13 | \( 1 + 5738 T + p^{7} T^{2} \) |
| 17 | \( 1 + 15654 T + p^{7} T^{2} \) |
| 19 | \( 1 + 19660 T + p^{7} T^{2} \) |
| 23 | \( 1 - 28512 T + p^{7} T^{2} \) |
| 29 | \( 1 + 140190 T + p^{7} T^{2} \) |
| 31 | \( 1 + 291208 T + p^{7} T^{2} \) |
| 37 | \( 1 - 135046 T + p^{7} T^{2} \) |
| 41 | \( 1 + 804438 T + p^{7} T^{2} \) |
| 43 | \( 1 + 721268 T + p^{7} T^{2} \) |
| 47 | \( 1 - 802656 T + p^{7} T^{2} \) |
| 53 | \( 1 + 274098 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1969440 T + p^{7} T^{2} \) |
| 61 | \( 1 - 3179342 T + p^{7} T^{2} \) |
| 67 | \( 1 - 1363756 T + p^{7} T^{2} \) |
| 71 | \( 1 + 4389888 T + p^{7} T^{2} \) |
| 73 | \( 1 - 4278862 T + p^{7} T^{2} \) |
| 79 | \( 1 - 3851960 T + p^{7} T^{2} \) |
| 83 | \( 1 + 8532228 T + p^{7} T^{2} \) |
| 89 | \( 1 - 3733410 T + p^{7} T^{2} \) |
| 97 | \( 1 - 15686206 T + p^{7} T^{2} \) |
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
−11.54466325671254800946626902753, −10.47239485625555102008135803663, −9.738707612153965317710199227083, −8.810829885858433226760004161082, −7.24362342518220086223891729598, −6.56158683221332900726193061900, −5.37083606252771175974221740484, −3.60591274566273053994568132103, −2.22305473115297885947965598664, −0.29020086086419870088409563594,
0.29020086086419870088409563594, 2.22305473115297885947965598664, 3.60591274566273053994568132103, 5.37083606252771175974221740484, 6.56158683221332900726193061900, 7.24362342518220086223891729598, 8.810829885858433226760004161082, 9.738707612153965317710199227083, 10.47239485625555102008135803663, 11.54466325671254800946626902753