| L(s) = 1 | + 1.82·2-s + 3-s + 1.33·4-s + 1.82·6-s + 1.70·7-s − 1.21·8-s + 9-s + 2.33·11-s + 1.33·12-s + 3.82·13-s + 3.11·14-s − 4.88·16-s + 1.58·17-s + 1.82·18-s − 5.43·19-s + 1.70·21-s + 4.26·22-s + 0.375·23-s − 1.21·24-s + 6.98·26-s + 27-s + 2.27·28-s + 5·29-s + 7.59·31-s − 6.49·32-s + 2.33·33-s + 2.90·34-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.577·3-s + 0.667·4-s + 0.745·6-s + 0.645·7-s − 0.429·8-s + 0.333·9-s + 0.703·11-s + 0.385·12-s + 1.06·13-s + 0.833·14-s − 1.22·16-s + 0.385·17-s + 0.430·18-s − 1.24·19-s + 0.372·21-s + 0.908·22-s + 0.0783·23-s − 0.248·24-s + 1.37·26-s + 0.192·27-s + 0.430·28-s + 0.928·29-s + 1.36·31-s − 1.14·32-s + 0.406·33-s + 0.497·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.161300339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.161300339\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 1.82T + 2T^{2} \) |
| 7 | \( 1 - 1.70T + 7T^{2} \) |
| 11 | \( 1 - 2.33T + 11T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 - 1.58T + 17T^{2} \) |
| 19 | \( 1 + 5.43T + 19T^{2} \) |
| 23 | \( 1 - 0.375T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 7.59T + 31T^{2} \) |
| 37 | \( 1 - 6.22T + 37T^{2} \) |
| 41 | \( 1 - 4.66T + 41T^{2} \) |
| 43 | \( 1 + 0.747T + 43T^{2} \) |
| 53 | \( 1 + 6.19T + 53T^{2} \) |
| 59 | \( 1 - 6.53T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 + 3.94T + 67T^{2} \) |
| 71 | \( 1 - 3.78T + 71T^{2} \) |
| 73 | \( 1 + 9.40T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 + 7.62T + 83T^{2} \) |
| 89 | \( 1 + 5.08T + 89T^{2} \) |
| 97 | \( 1 + 5.54T + 97T^{2} \) |
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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
−8.441270215380408089582505704594, −7.996025182967901141457855177350, −6.69188953358852619540727890769, −6.33959129439002209921057562701, −5.44041643848483301544225088598, −4.44818827577493141292306291539, −4.13217414832931042431673894736, −3.20234684809325395012736556619, −2.36040449207152085604723027132, −1.17993749177348907224941681853,
1.17993749177348907224941681853, 2.36040449207152085604723027132, 3.20234684809325395012736556619, 4.13217414832931042431673894736, 4.44818827577493141292306291539, 5.44041643848483301544225088598, 6.33959129439002209921057562701, 6.69188953358852619540727890769, 7.996025182967901141457855177350, 8.441270215380408089582505704594
