| L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s + 2·7-s + 8·8-s + 10·10-s + 13·13-s + 4·14-s + 16·16-s + 60·17-s + 50·19-s + 20·20-s − 210·23-s + 25·25-s + 26·26-s + 8·28-s + 228·29-s + 116·31-s + 32·32-s + 120·34-s + 10·35-s + 386·37-s + 100·38-s + 40·40-s − 378·41-s − 4·43-s − 420·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.107·7-s + 0.353·8-s + 0.316·10-s + 0.277·13-s + 0.0763·14-s + 1/4·16-s + 0.856·17-s + 0.603·19-s + 0.223·20-s − 1.90·23-s + 1/5·25-s + 0.196·26-s + 0.0539·28-s + 1.45·29-s + 0.672·31-s + 0.176·32-s + 0.605·34-s + 0.0482·35-s + 1.71·37-s + 0.426·38-s + 0.158·40-s − 1.43·41-s − 0.0141·43-s − 1.34·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.112104789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.112104789\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 - 60 T + p^{3} T^{2} \) |
| 19 | \( 1 - 50 T + p^{3} T^{2} \) |
| 23 | \( 1 + 210 T + p^{3} T^{2} \) |
| 29 | \( 1 - 228 T + p^{3} T^{2} \) |
| 31 | \( 1 - 116 T + p^{3} T^{2} \) |
| 37 | \( 1 - 386 T + p^{3} T^{2} \) |
| 41 | \( 1 + 378 T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 T + p^{3} T^{2} \) |
| 47 | \( 1 - 312 T + p^{3} T^{2} \) |
| 53 | \( 1 - 198 T + p^{3} T^{2} \) |
| 59 | \( 1 + 624 T + p^{3} T^{2} \) |
| 61 | \( 1 - 638 T + p^{3} T^{2} \) |
| 67 | \( 1 - 200 T + p^{3} T^{2} \) |
| 71 | \( 1 - 408 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1148 T + p^{3} T^{2} \) |
| 79 | \( 1 - 824 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1332 T + p^{3} T^{2} \) |
| 89 | \( 1 + 54 T + p^{3} T^{2} \) |
| 97 | \( 1 + 244 T + p^{3} T^{2} \) |
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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
−9.710289586326277813332907918101, −8.370072939859242635598481966769, −7.80063865146715241808053472067, −6.64339872274938895980480050397, −5.99356670277130883636764689131, −5.16294208068740420466698992566, −4.21603112034952563277537868141, −3.21818765981776693881851225469, −2.18369675082769213176863765955, −0.983333561891427777611974246968,
0.983333561891427777611974246968, 2.18369675082769213176863765955, 3.21818765981776693881851225469, 4.21603112034952563277537868141, 5.16294208068740420466698992566, 5.99356670277130883636764689131, 6.64339872274938895980480050397, 7.80063865146715241808053472067, 8.370072939859242635598481966769, 9.710289586326277813332907918101
