| L(s) = 1 | + (0.5 + 0.866i)5-s + (−0.366 + 0.366i)7-s + i·9-s + i·11-s + (−0.366 − 0.366i)17-s − 19-s + (−1 − i)23-s + (−0.499 + 0.866i)25-s + (−0.5 − 0.133i)35-s + (1.36 + 1.36i)43-s + (−0.866 + 0.5i)45-s + (1.36 − 1.36i)47-s + 0.732i·49-s + (−0.866 + 0.5i)55-s + 1.73·61-s + ⋯ |
| L(s) = 1 | + (0.5 + 0.866i)5-s + (−0.366 + 0.366i)7-s + i·9-s + i·11-s + (−0.366 − 0.366i)17-s − 19-s + (−1 − i)23-s + (−0.499 + 0.866i)25-s + (−0.5 − 0.133i)35-s + (1.36 + 1.36i)43-s + (−0.866 + 0.5i)45-s + (1.36 − 1.36i)47-s + 0.732i·49-s + (−0.866 + 0.5i)55-s + 1.73·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0299 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.027952707\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.027952707\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - iT^{2} \) |
| 7 | \( 1 + (0.366 - 0.366i)T - iT^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 47 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.73T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + (-1 - i)T + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.952342543339340078308453135336, −9.195287843303957661920712544942, −8.211697489727363970235403676280, −7.38706915222616316747542392671, −6.61549018060701743985103103810, −5.90928448791373290753344719820, −4.88229295106553967430353283597, −3.98211152832486381040168214490, −2.50676064991783948939205300226, −2.15321984517466962522476905279,
0.811533867645804144542630610230, 2.21508421632310889293862330603, 3.63995718415112392048800341318, 4.21440093897340253022766992983, 5.56986907657189080524550175164, 6.05854188676494843219046877587, 6.90444707531594704087674833851, 8.053536682626320029754828725457, 8.778431526336119632838472261702, 9.346615549358828817863754209281
