| L(s) = 1 | + (2.18 + 0.456i)5-s − 0.913i·7-s − 3.58i·11-s − 0.913·13-s + 3.58i·17-s + 4i·19-s + (4.58 + 1.99i)25-s + 7.84i·29-s + 5.29·31-s + (0.417 − 1.99i)35-s + 7.84·37-s − 6·41-s + 7.16·43-s − 6.92i·47-s + 6.16·49-s + ⋯ |
| L(s) = 1 | + (0.978 + 0.204i)5-s − 0.345i·7-s − 1.08i·11-s − 0.253·13-s + 0.868i·17-s + 0.917i·19-s + (0.916 + 0.399i)25-s + 1.45i·29-s + 0.950·31-s + (0.0705 − 0.338i)35-s + 1.28·37-s − 0.937·41-s + 1.09·43-s − 1.01i·47-s + 0.880·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.268768485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.268768485\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.18 - 0.456i)T \) |
| good | 7 | \( 1 + 0.913iT - 7T^{2} \) |
| 11 | \( 1 + 3.58iT - 11T^{2} \) |
| 13 | \( 1 + 0.913T + 13T^{2} \) |
| 17 | \( 1 - 3.58iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 7.84iT - 29T^{2} \) |
| 31 | \( 1 - 5.29T + 31T^{2} \) |
| 37 | \( 1 - 7.84T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 7.16T + 43T^{2} \) |
| 47 | \( 1 + 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 2.55T + 53T^{2} \) |
| 59 | \( 1 + 7.58iT - 59T^{2} \) |
| 61 | \( 1 + 10.5iT - 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 + 5.29T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 7.16iT - 97T^{2} \) |
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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
−8.677260088815837006794725604025, −8.201952063085393071315011257400, −7.16668952556725296125303522874, −6.39483583162784156318606680057, −5.79686551465000238396544483670, −5.06562931698241267759128698815, −3.91449068339197801132186917524, −3.11952704569113142957106461583, −2.06259382541611049206974723662, −0.992494177116205024785220714603,
0.925013063043467481159694749146, 2.30968050079228806184489651213, 2.67737363781925643105927537522, 4.29131852140292805260449966751, 4.85327604401776797194254740743, 5.70557830218082087940130344419, 6.45057261719862674051948952434, 7.21989849120818670034236225228, 7.974383112713288416894012136975, 9.038226326127539667675017350933
