L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 7-s + 8-s + 9-s − 11-s − 2·12-s − 2·13-s − 14-s + 16-s − 2·17-s + 18-s + 6·19-s + 2·21-s − 22-s − 6·23-s − 2·24-s − 2·26-s + 4·27-s − 28-s + 4·29-s + 32-s + 2·33-s − 2·34-s + 36-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.37·19-s + 0.436·21-s − 0.213·22-s − 1.25·23-s − 0.408·24-s − 0.392·26-s + 0.769·27-s − 0.188·28-s + 0.742·29-s + 0.176·32-s + 0.348·33-s − 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.458367802\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.458367802\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−8.364236767343812566058459791149, −7.47061829047941771853851611165, −6.78841724535451684686699565182, −6.15479195176174877530934120052, −5.37951608966004812570454158780, −4.98991370688772168201208158046, −4.00243555969074118936455684261, −3.09000934358695030931506608380, −2.09189918687403940235371222096, −0.64470210747492339700988718720,
0.64470210747492339700988718720, 2.09189918687403940235371222096, 3.09000934358695030931506608380, 4.00243555969074118936455684261, 4.98991370688772168201208158046, 5.37951608966004812570454158780, 6.15479195176174877530934120052, 6.78841724535451684686699565182, 7.47061829047941771853851611165, 8.364236767343812566058459791149