L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s + 9-s − 10-s + 5·11-s − 12-s + 2·14-s + 15-s + 16-s − 2·17-s + 18-s + 2·19-s − 20-s − 2·21-s + 5·22-s − 23-s − 24-s + 25-s − 27-s + 2·28-s + 5·29-s + 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s − 0.288·12-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.458·19-s − 0.223·20-s − 0.436·21-s + 1.06·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s + 0.928·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.003297385\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.003297385\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.292256770210165011885053358688, −7.24697168605499768849881069112, −6.66538797040504930804369683662, −6.14335060912400559351896538907, −5.11629200415445801955512861952, −4.58083529894544902329913632536, −3.95355379992984039564486269076, −3.05229051680019100251334236237, −1.82288921888896848678422549726, −0.935154422437708066127022887210,
0.935154422437708066127022887210, 1.82288921888896848678422549726, 3.05229051680019100251334236237, 3.95355379992984039564486269076, 4.58083529894544902329913632536, 5.11629200415445801955512861952, 6.14335060912400559351896538907, 6.66538797040504930804369683662, 7.24697168605499768849881069112, 8.292256770210165011885053358688