| L(s) = 1 | − 5·5-s − 32·7-s + 36·11-s − 10·13-s + 78·17-s − 140·19-s − 192·23-s + 25·25-s − 6·29-s + 16·31-s + 160·35-s − 34·37-s + 390·41-s + 52·43-s + 408·47-s + 681·49-s + 114·53-s − 180·55-s + 516·59-s − 58·61-s + 50·65-s + 892·67-s − 120·71-s − 646·73-s − 1.15e3·77-s + 1.16e3·79-s − 732·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.72·7-s + 0.986·11-s − 0.213·13-s + 1.11·17-s − 1.69·19-s − 1.74·23-s + 1/5·25-s − 0.0384·29-s + 0.0926·31-s + 0.772·35-s − 0.151·37-s + 1.48·41-s + 0.184·43-s + 1.26·47-s + 1.98·49-s + 0.295·53-s − 0.441·55-s + 1.13·59-s − 0.121·61-s + 0.0954·65-s + 1.62·67-s − 0.200·71-s − 1.03·73-s − 1.70·77-s + 1.66·79-s − 0.968·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.135901801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.135901801\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 140 T + p^{3} T^{2} \) |
| 23 | \( 1 + 192 T + p^{3} T^{2} \) |
| 29 | \( 1 + 6 T + p^{3} T^{2} \) |
| 31 | \( 1 - 16 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 390 T + p^{3} T^{2} \) |
| 43 | \( 1 - 52 T + p^{3} T^{2} \) |
| 47 | \( 1 - 408 T + p^{3} T^{2} \) |
| 53 | \( 1 - 114 T + p^{3} T^{2} \) |
| 59 | \( 1 - 516 T + p^{3} T^{2} \) |
| 61 | \( 1 + 58 T + p^{3} T^{2} \) |
| 67 | \( 1 - 892 T + p^{3} T^{2} \) |
| 71 | \( 1 + 120 T + p^{3} T^{2} \) |
| 73 | \( 1 + 646 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1168 T + p^{3} T^{2} \) |
| 83 | \( 1 + 732 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1590 T + p^{3} T^{2} \) |
| 97 | \( 1 - 2 p T + p^{3} T^{2} \) |
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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
−9.948943151524774269076606562536, −9.277064500079016034987917324073, −8.347613759734358918530732483072, −7.30205102656031344667386087787, −6.41117117781216744707866810815, −5.83717832399257337782806826057, −4.16243700642419100532598394699, −3.61923333518334604060585225870, −2.34996550677371331764509638286, −0.58865887783941126036912053374,
0.58865887783941126036912053374, 2.34996550677371331764509638286, 3.61923333518334604060585225870, 4.16243700642419100532598394699, 5.83717832399257337782806826057, 6.41117117781216744707866810815, 7.30205102656031344667386087787, 8.347613759734358918530732483072, 9.277064500079016034987917324073, 9.948943151524774269076606562536
