L(s) = 1 | + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (0.292 + 0.707i)11-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (−0.292 + 0.707i)29-s + (−1.70 + 0.707i)37-s + (0.707 + 1.70i)43-s − 1.00i·49-s + (0.707 + 1.70i)53-s + 1.00·63-s + (−0.707 + 1.70i)67-s + (−0.707 − 0.292i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)7-s + (−0.707 − 0.707i)9-s + (0.292 + 0.707i)11-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (−0.292 + 0.707i)29-s + (−1.70 + 0.707i)37-s + (0.707 + 1.70i)43-s − 1.00i·49-s + (0.707 + 1.70i)53-s + 1.00·63-s + (−0.707 + 1.70i)67-s + (−0.707 − 0.292i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9417996379\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9417996379\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 3 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 5 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 - iT^{2} \) |
| 43 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 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
−8.973140837725068775369346172277, −8.367536446704372837571398415693, −7.20804451460797334147010443232, −6.73332731206263269836249060569, −5.89318797626734875488982857651, −5.26601461054313002469683555138, −4.26690851059388906974143169980, −3.23205804989832047987685988740, −2.70305810637319658797469385740, −1.33818556008429459999772826554,
0.57184627605194205377383849911, 2.09369277184077846606514369400, 3.14739466895736509218870991284, 3.74922575953610651330772104277, 4.85305527769848657169341325760, 5.54905046090371344498692225880, 6.41308578932996328607184793168, 7.07700923915932512084401364385, 7.77752627077723301187017365798, 8.826664949840606702659203918090