L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (1.70 − 0.707i)7-s + (0.707 − 0.707i)8-s + (−1.70 − 0.707i)14-s − 1.00·16-s + i·17-s + (0.707 − 0.292i)23-s + (−0.707 − 0.707i)25-s + (0.707 + 1.70i)28-s + (−0.707 + 1.70i)31-s + (0.707 + 0.707i)32-s + (0.707 − 0.707i)34-s + (−0.707 − 1.70i)41-s + (−0.707 − 0.292i)46-s + 1.41·47-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + 1.00i·4-s + (1.70 − 0.707i)7-s + (0.707 − 0.707i)8-s + (−1.70 − 0.707i)14-s − 1.00·16-s + i·17-s + (0.707 − 0.292i)23-s + (−0.707 − 0.707i)25-s + (0.707 + 1.70i)28-s + (−0.707 + 1.70i)31-s + (0.707 + 0.707i)32-s + (0.707 − 0.707i)34-s + (−0.707 − 1.70i)41-s + (−0.707 − 0.292i)46-s + 1.41·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8964122017\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8964122017\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 17 | \( 1 - iT \) |
good | 5 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 7 | \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 23 | \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 29 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 79 | \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
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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
−10.03030208758591753728528364140, −8.771639571424078202412513634073, −8.419184524582442586021521033471, −7.54519920941229657910460169511, −6.89058256931300398219685241970, −5.39520611155340013230683026039, −4.42023273534952290825851311649, −3.66391248458661202563698091140, −2.18712915336866270881471041127, −1.27643285912743640047806963889,
1.41688532452101076396950036734, 2.47962352797793804133462338870, 4.32999604909393548255571015912, 5.23450734175194440850925135266, 5.72076260195449533797532406494, 6.97303473770196061275512829990, 7.75227467163108599076983390072, 8.260944905689909960277755999715, 9.190682963115202301233238665847, 9.678356416078570515628099127452