| L(s) = 1 | + 4·5-s − 3·9-s + 4·13-s − 2·17-s + 11·25-s + 4·29-s − 12·37-s − 10·41-s − 12·45-s − 7·49-s + 4·53-s − 12·61-s + 16·65-s − 6·73-s + 9·81-s − 8·85-s + 10·89-s − 18·97-s + 20·101-s + 20·109-s − 14·113-s − 12·117-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 9-s + 1.10·13-s − 0.485·17-s + 11/5·25-s + 0.742·29-s − 1.97·37-s − 1.56·41-s − 1.78·45-s − 49-s + 0.549·53-s − 1.53·61-s + 1.98·65-s − 0.702·73-s + 81-s − 0.867·85-s + 1.05·89-s − 1.82·97-s + 1.99·101-s + 1.91·109-s − 1.31·113-s − 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.559084749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.559084749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.01568276355264983627359007179, −10.87570847329036265183322640423, −10.16931220013108979582091687953, −9.056534895339973322073468947036, −8.475796264325709632552440745081, −6.69506256959746245491790982111, −5.96757132614070166013631000879, −5.05715565103777519658789739178, −3.14833606995625943162037597461, −1.77744920312998216430173831095,
1.77744920312998216430173831095, 3.14833606995625943162037597461, 5.05715565103777519658789739178, 5.96757132614070166013631000879, 6.69506256959746245491790982111, 8.475796264325709632552440745081, 9.056534895339973322073468947036, 10.16931220013108979582091687953, 10.87570847329036265183322640423, 12.01568276355264983627359007179
