| L(s) = 1 | − 2·3-s + 5·5-s + 12·7-s − 23·9-s + 20·11-s − 4·13-s − 10·15-s − 34·17-s + 19·19-s − 24·21-s − 40·23-s + 25·25-s + 100·27-s − 150·29-s + 200·31-s − 40·33-s + 60·35-s − 156·37-s + 8·39-s − 218·41-s − 248·43-s − 115·45-s + 180·47-s − 199·49-s + 68·51-s + 72·53-s + 100·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 0.447·5-s + 0.647·7-s − 0.851·9-s + 0.548·11-s − 0.0853·13-s − 0.172·15-s − 0.485·17-s + 0.229·19-s − 0.249·21-s − 0.362·23-s + 1/5·25-s + 0.712·27-s − 0.960·29-s + 1.15·31-s − 0.211·33-s + 0.289·35-s − 0.693·37-s + 0.0328·39-s − 0.830·41-s − 0.879·43-s − 0.380·45-s + 0.558·47-s − 0.580·49-s + 0.186·51-s + 0.186·53-s + 0.245·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 T + p^{3} T^{2} \) |
| 17 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 40 T + p^{3} T^{2} \) |
| 29 | \( 1 + 150 T + p^{3} T^{2} \) |
| 31 | \( 1 - 200 T + p^{3} T^{2} \) |
| 37 | \( 1 + 156 T + p^{3} T^{2} \) |
| 41 | \( 1 + 218 T + p^{3} T^{2} \) |
| 43 | \( 1 + 248 T + p^{3} T^{2} \) |
| 47 | \( 1 - 180 T + p^{3} T^{2} \) |
| 53 | \( 1 - 72 T + p^{3} T^{2} \) |
| 59 | \( 1 - 48 T + p^{3} T^{2} \) |
| 61 | \( 1 + 134 T + p^{3} T^{2} \) |
| 67 | \( 1 + 334 T + p^{3} T^{2} \) |
| 71 | \( 1 - 520 T + p^{3} T^{2} \) |
| 73 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 79 | \( 1 + 980 T + p^{3} T^{2} \) |
| 83 | \( 1 - 156 T + p^{3} T^{2} \) |
| 89 | \( 1 - 670 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1124 T + p^{3} T^{2} \) |
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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
−8.678715512127931700664413666330, −8.037480774390777631995598337691, −6.94052700316954273685712382336, −6.20944292793238626800354366228, −5.39392451923096282231365345709, −4.67728839305345690740549004031, −3.52880206926764412438047664380, −2.37969707264599612642460780845, −1.35134891655140811750696661930, 0,
1.35134891655140811750696661930, 2.37969707264599612642460780845, 3.52880206926764412438047664380, 4.67728839305345690740549004031, 5.39392451923096282231365345709, 6.20944292793238626800354366228, 6.94052700316954273685712382336, 8.037480774390777631995598337691, 8.678715512127931700664413666330
