L(s) = 1 | − 3-s − 3·7-s + 9-s − 2.87·11-s − 4.87·13-s + 3.87·17-s + 4.87·19-s + 3·21-s + 23-s − 27-s + 1.87·29-s + 3·31-s + 2.87·33-s − 37-s + 4.87·39-s + 1.87·41-s + 11.7·43-s − 0.872·47-s + 2·49-s − 3.87·51-s − 3.87·53-s − 4.87·57-s − 1.87·59-s + 1.12·61-s − 3·63-s + 4.74·67-s − 69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.13·7-s + 0.333·9-s − 0.866·11-s − 1.35·13-s + 0.939·17-s + 1.11·19-s + 0.654·21-s + 0.208·23-s − 0.192·27-s + 0.347·29-s + 0.538·31-s + 0.500·33-s − 0.164·37-s + 0.780·39-s + 0.292·41-s + 1.79·43-s − 0.127·47-s + 0.285·49-s − 0.542·51-s − 0.531·53-s − 0.645·57-s − 0.243·59-s + 0.144·61-s − 0.377·63-s + 0.579·67-s − 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 + 4.87T + 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 19 | \( 1 - 4.87T + 19T^{2} \) |
| 29 | \( 1 - 1.87T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 + T + 37T^{2} \) |
| 41 | \( 1 - 1.87T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 + 0.872T + 47T^{2} \) |
| 53 | \( 1 + 3.87T + 53T^{2} \) |
| 59 | \( 1 + 1.87T + 59T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 - 4.74T + 67T^{2} \) |
| 71 | \( 1 - 9.61T + 71T^{2} \) |
| 73 | \( 1 + 4.87T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 7.87T + 83T^{2} \) |
| 89 | \( 1 + 13.7T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−7.55647087150825578568342571058, −6.92187189444806396215679280610, −6.19257415179649549509327488827, −5.40212686275545553616041329411, −5.00613903467824252319079658300, −3.98073664328604602248110964222, −3.03230111305358924675300932909, −2.50915523538712413071140339163, −1.03530282582069993636146743868, 0,
1.03530282582069993636146743868, 2.50915523538712413071140339163, 3.03230111305358924675300932909, 3.98073664328604602248110964222, 5.00613903467824252319079658300, 5.40212686275545553616041329411, 6.19257415179649549509327488827, 6.92187189444806396215679280610, 7.55647087150825578568342571058