| L(s) = 1 | + 3-s + 9-s − 11-s + 8·19-s + 6·23-s + 27-s − 2·31-s − 33-s + 37-s − 6·41-s + 4·43-s − 7·49-s − 6·53-s + 8·57-s + 10·59-s + 12·61-s + 6·69-s − 16·71-s − 14·73-s + 8·79-s + 81-s − 4·83-s − 14·89-s − 2·93-s + 10·97-s − 99-s + 101-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.301·11-s + 1.83·19-s + 1.25·23-s + 0.192·27-s − 0.359·31-s − 0.174·33-s + 0.164·37-s − 0.937·41-s + 0.609·43-s − 49-s − 0.824·53-s + 1.05·57-s + 1.30·59-s + 1.53·61-s + 0.722·69-s − 1.89·71-s − 1.63·73-s + 0.900·79-s + 1/9·81-s − 0.439·83-s − 1.48·89-s − 0.207·93-s + 1.01·97-s − 0.100·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.610910898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.610910898\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−13.37604821747138, −13.19827056383078, −12.77246080352533, −12.02325505827683, −11.65296428116649, −11.16613372347675, −10.64569449272333, −9.914537249778401, −9.760597792004149, −9.134729318544819, −8.632416640723394, −8.205311996407883, −7.544476407178419, −7.139162507191618, −6.834079619748814, −5.835535940693234, −5.580158728841769, −4.794373695141436, −4.524572964189027, −3.484057273826683, −3.286102309838792, −2.696351425527709, −1.931852191998098, −1.264030773665160, −0.5869624225968860,
0.5869624225968860, 1.264030773665160, 1.931852191998098, 2.696351425527709, 3.286102309838792, 3.484057273826683, 4.524572964189027, 4.794373695141436, 5.580158728841769, 5.835535940693234, 6.834079619748814, 7.139162507191618, 7.544476407178419, 8.205311996407883, 8.632416640723394, 9.134729318544819, 9.760597792004149, 9.914537249778401, 10.64569449272333, 11.16613372347675, 11.65296428116649, 12.02325505827683, 12.77246080352533, 13.19827056383078, 13.37604821747138
