L(s) = 1 | + (1.20 + 1.08i)5-s + (0.913 − 0.406i)7-s + (−0.743 + 0.669i)11-s + (−1.58 + 0.336i)13-s + (0.587 − 0.190i)17-s + (0.809 − 0.587i)19-s + (−0.535 + 0.309i)23-s + (0.169 + 1.60i)25-s + (0.406 + 0.913i)29-s + (0.978 − 0.207i)31-s + (1.53 + 0.5i)35-s + (−0.406 + 0.913i)41-s + (1.60 − 0.169i)47-s + (0.951 + 0.309i)53-s − 1.61·55-s + ⋯ |
L(s) = 1 | + (1.20 + 1.08i)5-s + (0.913 − 0.406i)7-s + (−0.743 + 0.669i)11-s + (−1.58 + 0.336i)13-s + (0.587 − 0.190i)17-s + (0.809 − 0.587i)19-s + (−0.535 + 0.309i)23-s + (0.169 + 1.60i)25-s + (0.406 + 0.913i)29-s + (0.978 − 0.207i)31-s + (1.53 + 0.5i)35-s + (−0.406 + 0.913i)41-s + (1.60 − 0.169i)47-s + (0.951 + 0.309i)53-s − 1.61·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.631321722\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.631321722\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (0.743 - 0.669i)T \) |
good | 5 | \( 1 + (-1.20 - 1.08i)T + (0.104 + 0.994i)T^{2} \) |
| 7 | \( 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| 13 | \( 1 + (1.58 - 0.336i)T + (0.913 - 0.406i)T^{2} \) |
| 17 | \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.535 - 0.309i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.406 - 0.913i)T + (-0.669 + 0.743i)T^{2} \) |
| 31 | \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.406 - 0.913i)T + (-0.669 - 0.743i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-1.60 + 0.169i)T + (0.978 - 0.207i)T^{2} \) |
| 53 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.994 - 0.104i)T + (0.978 + 0.207i)T^{2} \) |
| 61 | \( 1 + (1.58 + 0.336i)T + (0.913 + 0.406i)T^{2} \) |
| 67 | \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.104 + 0.994i)T^{2} \) |
| 83 | \( 1 + (-0.207 + 0.978i)T + (-0.913 - 0.406i)T^{2} \) |
| 89 | \( 1 + 1.61iT - T^{2} \) |
| 97 | \( 1 + (-0.104 + 0.994i)T^{2} \) |
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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
−9.021644368257751657892549691773, −7.76850976039614700918998175043, −7.39793764275116832380139218818, −6.78479237598821253851383337453, −5.78785428428177101893254343429, −5.06992007892369485241133775607, −4.48932934256757211535353368588, −3.00136535349787041710347863475, −2.46198265556161754559458097216, −1.53823913125503769556910097854,
1.00593121680618687894820889753, 2.11711789234987071313604964184, 2.76317115143506943659620097262, 4.24509441150959729683037838743, 5.14318336760150322650002158506, 5.43656517140220737673300598598, 6.06574383545018367238610621249, 7.34855180477631459583640949142, 8.056956744408854661083837535837, 8.554419734524566363617285548477