| L(s) = 1 | + 4·2-s + 9·3-s + 16·4-s − 46·5-s + 36·6-s + 2·7-s + 64·8-s + 81·9-s − 184·10-s − 142·11-s + 144·12-s + 8·14-s − 414·15-s + 256·16-s + 324·18-s − 736·20-s + 18·21-s − 568·22-s + 576·24-s + 1.49e3·25-s + 729·27-s + 32·28-s + 818·29-s − 1.65e3·30-s − 478·31-s + 1.02e3·32-s − 1.27e3·33-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s − 1.83·5-s + 6-s + 2/49·7-s + 8-s + 9-s − 1.83·10-s − 1.17·11-s + 12-s + 2/49·14-s − 1.83·15-s + 16-s + 18-s − 1.83·20-s + 2/49·21-s − 1.17·22-s + 24-s + 2.38·25-s + 27-s + 2/49·28-s + 0.972·29-s − 1.83·30-s − 0.497·31-s + 32-s − 1.17·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.335028738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.335028738\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 46 T + p^{4} T^{2} \) |
| 7 | \( 1 - 2 T + p^{4} T^{2} \) |
| 11 | \( 1 + 142 T + p^{4} T^{2} \) |
| 13 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 17 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 19 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 23 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 29 | \( 1 - 818 T + p^{4} T^{2} \) |
| 31 | \( 1 + 478 T + p^{4} T^{2} \) |
| 37 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 41 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 43 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 47 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 53 | \( 1 - 3218 T + p^{4} T^{2} \) |
| 59 | \( 1 + 6862 T + p^{4} T^{2} \) |
| 61 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 67 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 71 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 73 | \( 1 + 8158 T + p^{4} T^{2} \) |
| 79 | \( 1 + 9118 T + p^{4} T^{2} \) |
| 83 | \( 1 - 4178 T + p^{4} T^{2} \) |
| 89 | \( ( 1 - p^{2} T )( 1 + p^{2} T ) \) |
| 97 | \( 1 - 17282 T + p^{4} T^{2} \) |
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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
−16.13061739844574942374608053746, −15.52553600653001149312352243379, −14.58794898805384163364586759724, −13.14453570882897817144770143952, −12.06291661157346517593908112426, −10.66125506501239425620016500066, −8.217827898116444756953209801854, −7.29211554830065822788646935492, −4.52416493388249072896626219377, −3.11944715513006857530985496671,
3.11944715513006857530985496671, 4.52416493388249072896626219377, 7.29211554830065822788646935492, 8.217827898116444756953209801854, 10.66125506501239425620016500066, 12.06291661157346517593908112426, 13.14453570882897817144770143952, 14.58794898805384163364586759724, 15.52553600653001149312352243379, 16.13061739844574942374608053746
