| L(s) = 1 | + 5·5-s − 6·7-s + 60·11-s + 50·13-s + 30·17-s − 40·19-s + 178·23-s + 25·25-s − 166·29-s − 20·31-s − 30·35-s + 10·37-s + 250·41-s − 142·43-s + 214·47-s − 307·49-s − 490·53-s + 300·55-s − 800·59-s + 250·61-s + 250·65-s + 774·67-s + 100·71-s − 230·73-s − 360·77-s + 1.32e3·79-s + 982·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.323·7-s + 1.64·11-s + 1.06·13-s + 0.428·17-s − 0.482·19-s + 1.61·23-s + 1/5·25-s − 1.06·29-s − 0.115·31-s − 0.144·35-s + 0.0444·37-s + 0.952·41-s − 0.503·43-s + 0.664·47-s − 0.895·49-s − 1.26·53-s + 0.735·55-s − 1.76·59-s + 0.524·61-s + 0.477·65-s + 1.41·67-s + 0.167·71-s − 0.368·73-s − 0.532·77-s + 1.87·79-s + 1.29·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.864847915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.864847915\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| good | 7 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 + 40 T + p^{3} T^{2} \) |
| 23 | \( 1 - 178 T + p^{3} T^{2} \) |
| 29 | \( 1 + 166 T + p^{3} T^{2} \) |
| 31 | \( 1 + 20 T + p^{3} T^{2} \) |
| 37 | \( 1 - 10 T + p^{3} T^{2} \) |
| 41 | \( 1 - 250 T + p^{3} T^{2} \) |
| 43 | \( 1 + 142 T + p^{3} T^{2} \) |
| 47 | \( 1 - 214 T + p^{3} T^{2} \) |
| 53 | \( 1 + 490 T + p^{3} T^{2} \) |
| 59 | \( 1 + 800 T + p^{3} T^{2} \) |
| 61 | \( 1 - 250 T + p^{3} T^{2} \) |
| 67 | \( 1 - 774 T + p^{3} T^{2} \) |
| 71 | \( 1 - 100 T + p^{3} T^{2} \) |
| 73 | \( 1 + 230 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1320 T + p^{3} T^{2} \) |
| 83 | \( 1 - 982 T + p^{3} T^{2} \) |
| 89 | \( 1 + 874 T + p^{3} T^{2} \) |
| 97 | \( 1 + 310 T + p^{3} T^{2} \) |
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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
−9.220303329146240259859344565328, −8.578242581569882216011601713494, −7.46395998388308491047528117431, −6.51593995579932503461637950273, −6.11218031408592648002839226550, −5.00377762129480538436152638122, −3.91281978276633002669522026210, −3.20050481210419138531977425851, −1.77711500518869572224339117811, −0.896322840759454357833538723908,
0.896322840759454357833538723908, 1.77711500518869572224339117811, 3.20050481210419138531977425851, 3.91281978276633002669522026210, 5.00377762129480538436152638122, 6.11218031408592648002839226550, 6.51593995579932503461637950273, 7.46395998388308491047528117431, 8.578242581569882216011601713494, 9.220303329146240259859344565328
