| L(s) = 1 | + (−1 + 2i)5-s − i·7-s + 4i·13-s + 4i·17-s − 4·19-s − 8i·23-s + (−3 − 4i)25-s + 2·29-s − 8·31-s + (2 + i)35-s + 8i·37-s − 6·41-s + 8i·43-s + 8i·47-s − 49-s + ⋯ |
| L(s) = 1 | + (−0.447 + 0.894i)5-s − 0.377i·7-s + 1.10i·13-s + 0.970i·17-s − 0.917·19-s − 1.66i·23-s + (−0.600 − 0.800i)25-s + 0.371·29-s − 1.43·31-s + (0.338 + 0.169i)35-s + 1.31i·37-s − 0.937·41-s + 1.21i·43-s + 1.16i·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6517733236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6517733236\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 - 2i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−10.28630686549445911423429436399, −9.181292137054726356015120717343, −8.345806656197979186835781356270, −7.58758503190917037834068109774, −6.50126589139299238862603764220, −6.35548679544745067335869699082, −4.67198866454970854600655997116, −4.03685911738431397278418732169, −2.96765216725405715849112457162, −1.77944218236695912773050263148,
0.26379374275432735683046553226, 1.79452251166668040447689198556, 3.18718983780593793711955095159, 4.12284037305519620721569663355, 5.30863946361300690156724032679, 5.62085455269324287305993389993, 7.08569018465165700243735778601, 7.72514320118517489118670067005, 8.643470355764417466753512147604, 9.165611770956031616213594010138
