L(s) = 1 | + 3-s + 4-s + 9-s + 12-s − 13-s + 16-s − 19-s + 27-s + 2·31-s + 36-s − 37-s − 39-s + 2·43-s + 48-s − 52-s − 57-s − 61-s + 64-s − 67-s − 73-s − 76-s − 79-s + 81-s + 2·93-s − 97-s − 103-s + 108-s + ⋯ |
L(s) = 1 | + 3-s + 4-s + 9-s + 12-s − 13-s + 16-s − 19-s + 27-s + 2·31-s + 36-s − 37-s − 39-s + 2·43-s + 48-s − 52-s − 57-s − 61-s + 64-s − 67-s − 73-s − 76-s − 79-s + 81-s + 2·93-s − 97-s − 103-s + 108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.310398290\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.310398290\) |
\(L(1)\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )( 1 + T ) \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )^{2} \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−8.612854364455431573638877889724, −7.911312816681841008682237773342, −7.31353228858888782389169738104, −6.67432974703489354095111753859, −5.90631666979448685601613484168, −4.75590359570656221553028693794, −4.01407498092970630582890012504, −2.87223491442559947554403010392, −2.46991844118178589201030165692, −1.43907181740291171896234292050,
1.43907181740291171896234292050, 2.46991844118178589201030165692, 2.87223491442559947554403010392, 4.01407498092970630582890012504, 4.75590359570656221553028693794, 5.90631666979448685601613484168, 6.67432974703489354095111753859, 7.31353228858888782389169738104, 7.911312816681841008682237773342, 8.612854364455431573638877889724