L(s) = 1 | + 3-s + 0.138·5-s − 3.33·7-s + 9-s − 2.06·11-s − 0.702·13-s + 0.138·15-s + 4.32·17-s + 2.44·19-s − 3.33·21-s + 23-s − 4.98·25-s + 27-s + 29-s + 7.41·31-s − 2.06·33-s − 0.462·35-s − 2.90·37-s − 0.702·39-s − 4.05·41-s + 0.235·43-s + 0.138·45-s − 2.05·47-s + 4.13·49-s + 4.32·51-s − 3.22·53-s − 0.286·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.0619·5-s − 1.26·7-s + 0.333·9-s − 0.622·11-s − 0.194·13-s + 0.0357·15-s + 1.04·17-s + 0.560·19-s − 0.728·21-s + 0.208·23-s − 0.996·25-s + 0.192·27-s + 0.185·29-s + 1.33·31-s − 0.359·33-s − 0.0781·35-s − 0.477·37-s − 0.112·39-s − 0.633·41-s + 0.0358·43-s + 0.0206·45-s − 0.299·47-s + 0.590·49-s + 0.605·51-s − 0.442·53-s − 0.0386·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 5 | \( 1 - 0.138T + 5T^{2} \) |
| 7 | \( 1 + 3.33T + 7T^{2} \) |
| 11 | \( 1 + 2.06T + 11T^{2} \) |
| 13 | \( 1 + 0.702T + 13T^{2} \) |
| 17 | \( 1 - 4.32T + 17T^{2} \) |
| 19 | \( 1 - 2.44T + 19T^{2} \) |
| 31 | \( 1 - 7.41T + 31T^{2} \) |
| 37 | \( 1 + 2.90T + 37T^{2} \) |
| 41 | \( 1 + 4.05T + 41T^{2} \) |
| 43 | \( 1 - 0.235T + 43T^{2} \) |
| 47 | \( 1 + 2.05T + 47T^{2} \) |
| 53 | \( 1 + 3.22T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 2.78T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 - 7.47T + 71T^{2} \) |
| 73 | \( 1 - 9.16T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 3.05T + 89T^{2} \) |
| 97 | \( 1 - 2.74T + 97T^{2} \) |
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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
−7.69676506297676817330748687948, −6.77175069762353648354717574600, −6.20758566833848920578150854031, −5.42069719547119284129037435638, −4.66458323920941180598129425459, −3.61957744589759668635710845991, −3.14503774895373652153171867080, −2.45492843636136302291258141979, −1.28444693408459198439036465212, 0,
1.28444693408459198439036465212, 2.45492843636136302291258141979, 3.14503774895373652153171867080, 3.61957744589759668635710845991, 4.66458323920941180598129425459, 5.42069719547119284129037435638, 6.20758566833848920578150854031, 6.77175069762353648354717574600, 7.69676506297676817330748687948