L(s) = 1 | + (1.86 − 0.604i)2-s + (−1.32 − 1.32i)3-s + (2.28 − 1.66i)4-s + (−3.25 − 1.65i)6-s + (2.10 − 2.89i)8-s + 2.48i·9-s + (−5.21 − 0.825i)12-s + (−0.571 + 1.12i)13-s + (1.28 − 3.96i)16-s + (1.50 + 4.62i)18-s + (0.309 + 0.951i)23-s + (−6.59 + 1.04i)24-s + (−0.309 + 0.951i)25-s + (−0.385 + 2.43i)26-s + (1.96 − 1.96i)27-s + ⋯ |
L(s) = 1 | + (1.86 − 0.604i)2-s + (−1.32 − 1.32i)3-s + (2.28 − 1.66i)4-s + (−3.25 − 1.65i)6-s + (2.10 − 2.89i)8-s + 2.48i·9-s + (−5.21 − 0.825i)12-s + (−0.571 + 1.12i)13-s + (1.28 − 3.96i)16-s + (1.50 + 4.62i)18-s + (0.309 + 0.951i)23-s + (−6.59 + 1.04i)24-s + (−0.309 + 0.951i)25-s + (−0.385 + 2.43i)26-s + (1.96 − 1.96i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 943 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.959933495\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.959933495\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (0.978 + 0.207i)T \) |
good | 2 | \( 1 + (-1.86 + 0.604i)T + (0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (1.32 + 1.32i)T + iT^{2} \) |
| 5 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 11 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 13 | \( 1 + (0.571 - 1.12i)T + (-0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 19 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (0.707 + 0.112i)T + (0.951 + 0.309i)T^{2} \) |
| 31 | \( 1 + (0.658 + 0.478i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 47 | \( 1 + (-1.77 - 0.906i)T + (0.587 + 0.809i)T^{2} \) |
| 53 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 59 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 71 | \( 1 + (0.262 + 1.65i)T + (-0.951 + 0.309i)T^{2} \) |
| 73 | \( 1 - 1.98iT - T^{2} \) |
| 79 | \( 1 - iT^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 97 | \( 1 + (0.951 + 0.309i)T^{2} \) |
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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
−10.64582709644350761263603335528, −9.516512303351855381940013001279, −7.48573986595846427270272878087, −7.09972216039789469121181019172, −6.24515683193038874259760283159, −5.52388577418529395086012055696, −4.91879066329061849432019344409, −3.78794485907801839933770364830, −2.27492893653081643326028451790, −1.50679053844028745711695509570,
2.79332280288089708555501706200, 3.83112923950824454397766459751, 4.53638309962611994332401125096, 5.31640766189241886448822414583, 5.79842836100730898769720272284, 6.64278942039344472127891849671, 7.54514419274586473982760684799, 8.828497448222521042703766766369, 10.32441845687406434829785089765, 10.62651815735479055154699997593