L(s) = 1 | − 2-s − 3-s + 4-s − 3·5-s + 6-s − 8-s + 9-s + 3·10-s + 3·11-s − 12-s + 4·13-s + 3·15-s + 16-s − 18-s + 4·19-s − 3·20-s − 3·22-s + 24-s + 4·25-s − 4·26-s − 27-s + 9·29-s − 3·30-s + 31-s − 32-s − 3·33-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.904·11-s − 0.288·12-s + 1.10·13-s + 0.774·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.670·20-s − 0.639·22-s + 0.204·24-s + 4/5·25-s − 0.784·26-s − 0.192·27-s + 1.67·29-s − 0.547·30-s + 0.179·31-s − 0.176·32-s − 0.522·33-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6693039983\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6693039983\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−11.68014818696489555780067605703, −11.01449612120722930985164678406, −9.959946615270396196728666018490, −8.804654554318459390164507311566, −8.000167698268580301677238135628, −7.00728426330403345958805784350, −6.07864651997567107362244436260, −4.49123641406362856214796748304, −3.36779260852424710508563371535, −1.01560111630863456380812991130,
1.01560111630863456380812991130, 3.36779260852424710508563371535, 4.49123641406362856214796748304, 6.07864651997567107362244436260, 7.00728426330403345958805784350, 8.000167698268580301677238135628, 8.804654554318459390164507311566, 9.959946615270396196728666018490, 11.01449612120722930985164678406, 11.68014818696489555780067605703