L(s) = 1 | + (0.873 − 1.20i)3-s + (0.5 − 0.866i)5-s + (−0.373 − 1.14i)9-s + (−0.309 + 0.951i)11-s + (−0.604 − 1.35i)15-s + (0.773 + 0.251i)23-s + (−0.499 − 0.866i)25-s + (−0.295 − 0.0960i)27-s + (−1.58 + 1.14i)31-s + (0.873 + 1.20i)33-s + (−1.64 + 0.535i)37-s + (−1.18 − 0.251i)45-s − 49-s + (1.11 − 1.53i)53-s + (0.669 + 0.743i)55-s + ⋯ |
L(s) = 1 | + (0.873 − 1.20i)3-s + (0.5 − 0.866i)5-s + (−0.373 − 1.14i)9-s + (−0.309 + 0.951i)11-s + (−0.604 − 1.35i)15-s + (0.773 + 0.251i)23-s + (−0.499 − 0.866i)25-s + (−0.295 − 0.0960i)27-s + (−1.58 + 1.14i)31-s + (0.873 + 1.20i)33-s + (−1.64 + 0.535i)37-s + (−1.18 − 0.251i)45-s − 49-s + (1.11 − 1.53i)53-s + (0.669 + 0.743i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.146 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.438657381\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.438657381\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.309 - 0.951i)T \) |
good | 3 | \( 1 + (-0.873 + 1.20i)T + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.773 - 0.251i)T + (0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (1.58 - 1.14i)T + (0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (1.64 - 0.535i)T + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.413 - 1.27i)T + (-0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-1.16 - 1.60i)T + (-0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (1.08 + 0.786i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.0646 + 0.198i)T + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.244 + 0.336i)T + (-0.309 - 0.951i)T^{2} \) |
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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
−9.667455645908098195051830171076, −8.790268675154382028105064005485, −8.377828544860265031975220234809, −7.26426483510555902009738322112, −6.90395988957023058371163930938, −5.58628650757131943795466055669, −4.78663062930310487871220596758, −3.39500040508258854286872999824, −2.17867466567828579123617630787, −1.43720025654986251813518532455,
2.22164132887692981900178327870, 3.21485288943417913671195187452, 3.78533184985689360915196138135, 5.05362137638548325051158275845, 5.88032093367915281448843133370, 6.96508600781463108751489802529, 7.924793122821666221360233492859, 8.874659297894971893372762752528, 9.367227912628026807586660675391, 10.24197594571595717584005867703