L(s) = 1 | + 2·29-s + 2·41-s + 49-s − 2·61-s + 2·89-s − 2·101-s + 2·109-s + ⋯ |
L(s) = 1 | + 2·29-s + 2·41-s + 49-s − 2·61-s + 2·89-s − 2·101-s + 2·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.306934344\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.306934344\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 + T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.765174895244531193974508901748, −7.964501655175975547360355392106, −7.32128235563975340193434195132, −6.43913369791821294170616881749, −5.84346598347673950005578099283, −4.84522398827095556756981450152, −4.21012682675798903946459521558, −3.15039301255661866883959383893, −2.33208052629464066896006072670, −1.03180259769072802093206375626,
1.03180259769072802093206375626, 2.33208052629464066896006072670, 3.15039301255661866883959383893, 4.21012682675798903946459521558, 4.84522398827095556756981450152, 5.84346598347673950005578099283, 6.43913369791821294170616881749, 7.32128235563975340193434195132, 7.964501655175975547360355392106, 8.765174895244531193974508901748