| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 11-s − 5·13-s + 15-s − 4·17-s − 5·19-s − 21-s − 6·23-s − 4·25-s + 27-s − 9·29-s + 4·31-s − 33-s − 35-s + 5·37-s − 5·39-s + 6·43-s + 45-s + 7·47-s + 49-s − 4·51-s + 2·53-s − 55-s − 5·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.38·13-s + 0.258·15-s − 0.970·17-s − 1.14·19-s − 0.218·21-s − 1.25·23-s − 4/5·25-s + 0.192·27-s − 1.67·29-s + 0.718·31-s − 0.174·33-s − 0.169·35-s + 0.821·37-s − 0.800·39-s + 0.914·43-s + 0.149·45-s + 1.02·47-s + 1/7·49-s − 0.560·51-s + 0.274·53-s − 0.134·55-s − 0.662·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.043455534841809898271245691483, −7.976421595815454413549588476837, −7.43514558588759140727172928796, −6.44032312459493558621376813099, −5.73557397507853566774798311107, −4.59009510878056701616277525037, −3.88389515680339763362894875056, −2.51755031434269220568444086289, −2.06472965316638953016723909481, 0,
2.06472965316638953016723909481, 2.51755031434269220568444086289, 3.88389515680339763362894875056, 4.59009510878056701616277525037, 5.73557397507853566774798311107, 6.44032312459493558621376813099, 7.43514558588759140727172928796, 7.976421595815454413549588476837, 9.043455534841809898271245691483
