| L(s) = 1 | + 3-s − 0.681·5-s + 3.18·7-s + 9-s + 3.18·11-s + 1.49·13-s − 0.681·15-s − 1.18·17-s − 19-s + 3.18·21-s + 4.85·23-s − 4.53·25-s + 27-s + 8.37·29-s − 7.23·31-s + 3.18·33-s − 2.17·35-s + 1.49·37-s + 1.49·39-s − 3.36·41-s − 1.82·43-s − 0.681·45-s + 1.66·47-s + 3.17·49-s − 1.18·51-s + 12.7·53-s − 2.17·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.304·5-s + 1.20·7-s + 0.333·9-s + 0.961·11-s + 0.413·13-s − 0.175·15-s − 0.288·17-s − 0.229·19-s + 0.696·21-s + 1.01·23-s − 0.907·25-s + 0.192·27-s + 1.55·29-s − 1.29·31-s + 0.555·33-s − 0.367·35-s + 0.245·37-s + 0.238·39-s − 0.525·41-s − 0.278·43-s − 0.101·45-s + 0.242·47-s + 0.453·49-s − 0.166·51-s + 1.74·53-s − 0.293·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.554739983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.554739983\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 0.681T + 5T^{2} \) |
| 7 | \( 1 - 3.18T + 7T^{2} \) |
| 11 | \( 1 - 3.18T + 11T^{2} \) |
| 13 | \( 1 - 1.49T + 13T^{2} \) |
| 17 | \( 1 + 1.18T + 17T^{2} \) |
| 23 | \( 1 - 4.85T + 23T^{2} \) |
| 29 | \( 1 - 8.37T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 - 1.49T + 37T^{2} \) |
| 41 | \( 1 + 3.36T + 41T^{2} \) |
| 43 | \( 1 + 1.82T + 43T^{2} \) |
| 47 | \( 1 - 1.66T + 47T^{2} \) |
| 53 | \( 1 - 12.7T + 53T^{2} \) |
| 59 | \( 1 - 4.34T + 59T^{2} \) |
| 61 | \( 1 + 2.55T + 61T^{2} \) |
| 67 | \( 1 + 9.36T + 67T^{2} \) |
| 71 | \( 1 + 4.34T + 71T^{2} \) |
| 73 | \( 1 - 5.53T + 73T^{2} \) |
| 79 | \( 1 + 8.24T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−8.863610340791652382944430730072, −8.694071138153838285990483373817, −7.73442617824641455865092999143, −7.06806715016470383192350919342, −6.12418269957408381711678124821, −5.00697943617052551020751261164, −4.25228537545059367925802902962, −3.44024832201114660808782879349, −2.16794563669697272490400730529, −1.18032222196348930112380877972,
1.18032222196348930112380877972, 2.16794563669697272490400730529, 3.44024832201114660808782879349, 4.25228537545059367925802902962, 5.00697943617052551020751261164, 6.12418269957408381711678124821, 7.06806715016470383192350919342, 7.73442617824641455865092999143, 8.694071138153838285990483373817, 8.863610340791652382944430730072
