L(s) = 1 | + 2.01·3-s − 5-s + 1.12·7-s + 1.06·9-s + 1.62·11-s − 5.08·13-s − 2.01·15-s + 0.834·17-s − 2.00·19-s + 2.26·21-s − 2.97·23-s + 25-s − 3.90·27-s + 7.67·29-s + 9.19·31-s + 3.27·33-s − 1.12·35-s − 6.56·37-s − 10.2·39-s − 11.5·41-s − 2.96·43-s − 1.06·45-s − 11.2·47-s − 5.73·49-s + 1.68·51-s − 1.53·53-s − 1.62·55-s + ⋯ |
L(s) = 1 | + 1.16·3-s − 0.447·5-s + 0.424·7-s + 0.354·9-s + 0.489·11-s − 1.41·13-s − 0.520·15-s + 0.202·17-s − 0.458·19-s + 0.493·21-s − 0.620·23-s + 0.200·25-s − 0.750·27-s + 1.42·29-s + 1.65·31-s + 0.570·33-s − 0.189·35-s − 1.07·37-s − 1.64·39-s − 1.80·41-s − 0.451·43-s − 0.158·45-s − 1.64·47-s − 0.819·49-s + 0.235·51-s − 0.211·53-s − 0.219·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 2.01T + 3T^{2} \) |
| 7 | \( 1 - 1.12T + 7T^{2} \) |
| 11 | \( 1 - 1.62T + 11T^{2} \) |
| 13 | \( 1 + 5.08T + 13T^{2} \) |
| 17 | \( 1 - 0.834T + 17T^{2} \) |
| 19 | \( 1 + 2.00T + 19T^{2} \) |
| 23 | \( 1 + 2.97T + 23T^{2} \) |
| 29 | \( 1 - 7.67T + 29T^{2} \) |
| 31 | \( 1 - 9.19T + 31T^{2} \) |
| 37 | \( 1 + 6.56T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 2.96T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 1.53T + 53T^{2} \) |
| 59 | \( 1 + 2.89T + 59T^{2} \) |
| 61 | \( 1 + 6.38T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 7.85T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 7.45T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.74017989336909828989552751687, −6.84331148975867051567562055235, −6.38309181638511074419593579322, −5.04679672456844666669424653909, −4.71248476369471219529377970875, −3.74296966222128485937779082184, −3.07985040501476524712063202100, −2.34969140840042865614521976545, −1.50251394353462875842998383170, 0,
1.50251394353462875842998383170, 2.34969140840042865614521976545, 3.07985040501476524712063202100, 3.74296966222128485937779082184, 4.71248476369471219529377970875, 5.04679672456844666669424653909, 6.38309181638511074419593579322, 6.84331148975867051567562055235, 7.74017989336909828989552751687