| L(s) = 1 | − 5-s − 2·7-s + 3·11-s − 2·17-s + 19-s − 2·23-s + 25-s + 3·29-s − 3·31-s + 2·35-s − 5·41-s − 4·43-s + 8·47-s − 3·49-s − 2·53-s − 3·55-s − 3·59-s + 6·61-s − 10·67-s + 15·71-s − 14·73-s − 6·77-s − 8·79-s + 2·85-s − 89-s − 95-s − 16·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.904·11-s − 0.485·17-s + 0.229·19-s − 0.417·23-s + 1/5·25-s + 0.557·29-s − 0.538·31-s + 0.338·35-s − 0.780·41-s − 0.609·43-s + 1.16·47-s − 3/7·49-s − 0.274·53-s − 0.404·55-s − 0.390·59-s + 0.768·61-s − 1.22·67-s + 1.78·71-s − 1.63·73-s − 0.683·77-s − 0.900·79-s + 0.216·85-s − 0.105·89-s − 0.102·95-s − 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.373144451675077111823016766462, −7.45527800657238257862866313400, −6.74512992177894309550576258587, −6.19244472846110384957077655120, −5.20870925292776089372449631208, −4.23030242668308709998880346697, −3.59637101529424453960177108255, −2.67882606004534936184693117968, −1.40921516626677664086056376863, 0,
1.40921516626677664086056376863, 2.67882606004534936184693117968, 3.59637101529424453960177108255, 4.23030242668308709998880346697, 5.20870925292776089372449631208, 6.19244472846110384957077655120, 6.74512992177894309550576258587, 7.45527800657238257862866313400, 8.373144451675077111823016766462
