| L(s) = 1 | − 2·3-s + 3·7-s + 9-s − 2·17-s + 2·19-s − 6·21-s − 9·23-s + 4·27-s − 2·29-s + 3·31-s + 12·37-s + 7·41-s − 8·43-s + 11·47-s + 2·49-s + 4·51-s + 8·53-s − 4·57-s + 10·59-s − 10·61-s + 3·63-s + 10·67-s + 18·69-s + 8·71-s − 11·73-s − 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.13·7-s + 1/3·9-s − 0.485·17-s + 0.458·19-s − 1.30·21-s − 1.87·23-s + 0.769·27-s − 0.371·29-s + 0.538·31-s + 1.97·37-s + 1.09·41-s − 1.21·43-s + 1.60·47-s + 2/7·49-s + 0.560·51-s + 1.09·53-s − 0.529·57-s + 1.30·59-s − 1.28·61-s + 0.377·63-s + 1.22·67-s + 2.16·69-s + 0.949·71-s − 1.28·73-s − 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.519751188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.519751188\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−14.50642102304710, −14.11388509706390, −13.53936655242328, −12.98721725263928, −12.26367511000812, −11.85207359936748, −11.54671758591387, −10.99970031679919, −10.64238613847218, −9.924051693902071, −9.529387375402406, −8.624762936246615, −8.250069810910146, −7.655077348182285, −7.129109507571261, −6.350310393072294, −5.872364267920013, −5.509071122212781, −4.754651932553169, −4.354758699684850, −3.752252950187559, −2.632636164570244, −2.101077403653753, −1.198736560026434, −0.5185218187200159,
0.5185218187200159, 1.198736560026434, 2.101077403653753, 2.632636164570244, 3.752252950187559, 4.354758699684850, 4.754651932553169, 5.509071122212781, 5.872364267920013, 6.350310393072294, 7.129109507571261, 7.655077348182285, 8.250069810910146, 8.624762936246615, 9.529387375402406, 9.924051693902071, 10.64238613847218, 10.99970031679919, 11.54671758591387, 11.85207359936748, 12.26367511000812, 12.98721725263928, 13.53936655242328, 14.11388509706390, 14.50642102304710
