| L(s) = 1 | + 0.915·3-s + 1.84·5-s − 3.15·7-s − 2.16·9-s + 3.23·11-s + 0.662·13-s + 1.68·15-s + 17-s + 3.12·19-s − 2.89·21-s − 8.89·23-s − 1.59·25-s − 4.72·27-s − 7.93·29-s − 8.13·31-s + 2.96·33-s − 5.82·35-s + 8.26·37-s + 0.606·39-s − 8.91·41-s − 6.95·43-s − 3.98·45-s + 0.528·47-s + 2.98·49-s + 0.915·51-s + 13.1·53-s + 5.97·55-s + ⋯ |
| L(s) = 1 | + 0.528·3-s + 0.824·5-s − 1.19·7-s − 0.720·9-s + 0.975·11-s + 0.183·13-s + 0.436·15-s + 0.242·17-s + 0.716·19-s − 0.631·21-s − 1.85·23-s − 0.319·25-s − 0.909·27-s − 1.47·29-s − 1.46·31-s + 0.516·33-s − 0.985·35-s + 1.35·37-s + 0.0971·39-s − 1.39·41-s − 1.06·43-s − 0.594·45-s + 0.0771·47-s + 0.425·49-s + 0.128·51-s + 1.79·53-s + 0.805·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - 0.915T + 3T^{2} \) |
| 5 | \( 1 - 1.84T + 5T^{2} \) |
| 7 | \( 1 + 3.15T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 - 0.662T + 13T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 + 8.89T + 23T^{2} \) |
| 29 | \( 1 + 7.93T + 29T^{2} \) |
| 31 | \( 1 + 8.13T + 31T^{2} \) |
| 37 | \( 1 - 8.26T + 37T^{2} \) |
| 41 | \( 1 + 8.91T + 41T^{2} \) |
| 43 | \( 1 + 6.95T + 43T^{2} \) |
| 47 | \( 1 - 0.528T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 61 | \( 1 - 2.06T + 61T^{2} \) |
| 67 | \( 1 + 3.33T + 67T^{2} \) |
| 71 | \( 1 + 1.13T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 3.07T + 89T^{2} \) |
| 97 | \( 1 - 4.23T + 97T^{2} \) |
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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
−8.140558155717689700771430630950, −7.36524329374509421564826793321, −6.45671289100806245310924069961, −5.91581286370472793619213661800, −5.37635544612353938632602205797, −3.80765628330663892647255004149, −3.56859450179351665961422658058, −2.45128344908059588271171862670, −1.64834261361765959550628914810, 0,
1.64834261361765959550628914810, 2.45128344908059588271171862670, 3.56859450179351665961422658058, 3.80765628330663892647255004149, 5.37635544612353938632602205797, 5.91581286370472793619213661800, 6.45671289100806245310924069961, 7.36524329374509421564826793321, 8.140558155717689700771430630950
