| L(s) = 1 | + 2·3-s − 5-s + 7-s + 9-s + 4·13-s − 2·15-s − 4·19-s + 2·21-s + 25-s − 4·27-s + 6·29-s + 10·31-s − 35-s + 2·37-s + 8·39-s + 12·41-s − 4·43-s − 45-s − 6·47-s + 49-s − 6·53-s − 8·57-s + 6·59-s + 4·61-s + 63-s − 4·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.10·13-s − 0.516·15-s − 0.917·19-s + 0.436·21-s + 1/5·25-s − 0.769·27-s + 1.11·29-s + 1.79·31-s − 0.169·35-s + 0.328·37-s + 1.28·39-s + 1.87·41-s − 0.609·43-s − 0.149·45-s − 0.875·47-s + 1/7·49-s − 0.824·53-s − 1.05·57-s + 0.781·59-s + 0.512·61-s + 0.125·63-s − 0.496·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.084503239\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.084503239\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.16658875188011, −13.75007315561450, −13.26110175674601, −12.86669581677019, −12.10654177857599, −11.73212099523757, −11.05498703676537, −10.74597759862748, −10.04654900031833, −9.504423888252285, −8.907604604164741, −8.435719905121451, −8.109996990563655, −7.790250889088491, −6.944186905492108, −6.339669785757068, −5.985341073615335, −4.996558672240871, −4.487046536683132, −3.919053004563178, −3.383798390441318, −2.713659540417637, −2.249382431461887, −1.369934413498655, −0.6547276201977499,
0.6547276201977499, 1.369934413498655, 2.249382431461887, 2.713659540417637, 3.383798390441318, 3.919053004563178, 4.487046536683132, 4.996558672240871, 5.985341073615335, 6.339669785757068, 6.944186905492108, 7.790250889088491, 8.109996990563655, 8.435719905121451, 8.907604604164741, 9.504423888252285, 10.04654900031833, 10.74597759862748, 11.05498703676537, 11.73212099523757, 12.10654177857599, 12.86669581677019, 13.26110175674601, 13.75007315561450, 14.16658875188011
