L(s) = 1 | + 2-s − 2·3-s + 4-s − 2.73·5-s − 2·6-s + 8-s + 9-s − 2.73·10-s − 2.73·11-s − 2·12-s + 5.46·13-s + 5.46·15-s + 16-s + 18-s + 3.46·19-s − 2.73·20-s − 2.73·22-s − 2·23-s − 2·24-s + 2.46·25-s + 5.46·26-s + 4·27-s − 7.66·29-s + 5.46·30-s − 1.46·31-s + 32-s + 5.46·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s + 0.5·4-s − 1.22·5-s − 0.816·6-s + 0.353·8-s + 0.333·9-s − 0.863·10-s − 0.823·11-s − 0.577·12-s + 1.51·13-s + 1.41·15-s + 0.250·16-s + 0.235·18-s + 0.794·19-s − 0.610·20-s − 0.582·22-s − 0.417·23-s − 0.408·24-s + 0.492·25-s + 1.07·26-s + 0.769·27-s − 1.42·29-s + 0.997·30-s − 0.262·31-s + 0.176·32-s + 0.951·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + 2T + 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 11 | \( 1 + 2.73T + 11T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 7.66T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 - 4.92T + 37T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + 0.535T + 47T^{2} \) |
| 53 | \( 1 - 7.66T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 6.73T + 61T^{2} \) |
| 67 | \( 1 - 0.196T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 - 8.92T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 4.73T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.79592827273602933833804996934, −7.37049211954217764058573524974, −6.34539695193228943783254383080, −5.76236492837864860855716988318, −5.20788284114810811986425276123, −4.21803941925655274431191779851, −3.70982137288922512259367674243, −2.73257066438540463357652840204, −1.20760190856188987621636981800, 0,
1.20760190856188987621636981800, 2.73257066438540463357652840204, 3.70982137288922512259367674243, 4.21803941925655274431191779851, 5.20788284114810811986425276123, 5.76236492837864860855716988318, 6.34539695193228943783254383080, 7.37049211954217764058573524974, 7.79592827273602933833804996934