| L(s) = 1 | + 2·3-s + 2·5-s − 7-s + 9-s − 5·11-s + 6·13-s + 4·15-s − 2·17-s − 4·19-s − 2·21-s − 3·23-s − 25-s − 4·27-s + 7·29-s + 10·31-s − 10·33-s − 2·35-s + 12·39-s + 10·41-s + 4·43-s + 2·45-s + 6·47-s + 49-s − 4·51-s + 3·53-s − 10·55-s − 8·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.50·11-s + 1.66·13-s + 1.03·15-s − 0.485·17-s − 0.917·19-s − 0.436·21-s − 0.625·23-s − 1/5·25-s − 0.769·27-s + 1.29·29-s + 1.79·31-s − 1.74·33-s − 0.338·35-s + 1.92·39-s + 1.56·41-s + 0.609·43-s + 0.298·45-s + 0.875·47-s + 1/7·49-s − 0.560·51-s + 0.412·53-s − 1.34·55-s − 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.909981796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.909981796\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.93463013710154, −13.68522876495772, −13.16221588459334, −12.84612547363203, −12.22971988784853, −11.47452731047165, −10.83540712296314, −10.37158699795994, −10.15921644697497, −9.204476851865157, −9.142039410541864, −8.380888594512539, −8.045049816494892, −7.693247855249279, −6.687267669556594, −6.204978016075898, −5.904931686868564, −5.223229219522413, −4.298747585386728, −4.035195667477618, −3.056382716192067, −2.676896963440905, −2.275086759724910, −1.500469650241644, −0.5798397516615105,
0.5798397516615105, 1.500469650241644, 2.275086759724910, 2.676896963440905, 3.056382716192067, 4.035195667477618, 4.298747585386728, 5.223229219522413, 5.904931686868564, 6.204978016075898, 6.687267669556594, 7.693247855249279, 8.045049816494892, 8.380888594512539, 9.142039410541864, 9.204476851865157, 10.15921644697497, 10.37158699795994, 10.83540712296314, 11.47452731047165, 12.22971988784853, 12.84612547363203, 13.16221588459334, 13.68522876495772, 13.93463013710154
