L(s) = 1 | + (1.58 + 1.58i)5-s + (−3.23 + 3.23i)7-s + 4.57i·11-s + (4.23 + 4.23i)13-s + (−1.74 − 1.74i)17-s − 2.47i·19-s + (−2.82 + 2.82i)23-s + 5.00i·25-s + 5.99·29-s + 1.52·31-s − 10.2·35-s + (−2.23 + 2.23i)37-s − 7.07i·41-s + (−2.47 − 2.47i)43-s + (1.74 + 1.74i)47-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)5-s + (−1.22 + 1.22i)7-s + 1.37i·11-s + (1.17 + 1.17i)13-s + (−0.423 − 0.423i)17-s − 0.567i·19-s + (−0.589 + 0.589i)23-s + 1.00i·25-s + 1.11·29-s + 0.274·31-s − 1.72·35-s + (−0.367 + 0.367i)37-s − 1.10i·41-s + (−0.376 − 0.376i)43-s + (0.254 + 0.254i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.437928567\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.437928567\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.58 - 1.58i)T \) |
good | 7 | \( 1 + (3.23 - 3.23i)T - 7iT^{2} \) |
| 11 | \( 1 - 4.57iT - 11T^{2} \) |
| 13 | \( 1 + (-4.23 - 4.23i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.74 + 1.74i)T + 17iT^{2} \) |
| 19 | \( 1 + 2.47iT - 19T^{2} \) |
| 23 | \( 1 + (2.82 - 2.82i)T - 23iT^{2} \) |
| 29 | \( 1 - 5.99T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 + (2.23 - 2.23i)T - 37iT^{2} \) |
| 41 | \( 1 + 7.07iT - 41T^{2} \) |
| 43 | \( 1 + (2.47 + 2.47i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.74 - 1.74i)T + 47iT^{2} \) |
| 53 | \( 1 - 53iT^{2} \) |
| 59 | \( 1 - 1.08T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + (-1.52 + 1.52i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.6iT - 71T^{2} \) |
| 73 | \( 1 + (0.527 + 0.527i)T + 73iT^{2} \) |
| 79 | \( 1 + 14.4iT - 79T^{2} \) |
| 83 | \( 1 + (-1.08 + 1.08i)T - 83iT^{2} \) |
| 89 | \( 1 + 0.746T + 89T^{2} \) |
| 97 | \( 1 + (-1 + i)T - 97iT^{2} \) |
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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
−9.139553262311185344450559464119, −8.687430536449105509402198439496, −7.32072495280994640334094506870, −6.64857194981863199549228310370, −6.26806098250561972888676369794, −5.41933983341177365945855251952, −4.39701013315321977059731733927, −3.34334448309895500124209913969, −2.48028428941003334262385647635, −1.77689977644250628307402449092,
0.47214775554387725136258167074, 1.26536145643358677044102245038, 2.89246746512536524182660959051, 3.58399893534283187513040096404, 4.39006328553071957853147387370, 5.58191351803530167305161050427, 6.22522198296460857030327757558, 6.58158639382893272992174570617, 7.975447533636271598815808559003, 8.360357122379965836065849360024