| L(s) = 1 | − 2-s − 3-s + 4-s − 3.46·5-s + 6-s − 8-s + 9-s + 3.46·10-s + 11-s − 12-s − 2·13-s + 3.46·15-s + 16-s + 3.46·17-s − 18-s + 1.46·19-s − 3.46·20-s − 22-s − 6.92·23-s + 24-s + 6.99·25-s + 2·26-s − 27-s − 6·29-s − 3.46·30-s + 1.46·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.54·5-s + 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.09·10-s + 0.301·11-s − 0.288·12-s − 0.554·13-s + 0.894·15-s + 0.250·16-s + 0.840·17-s − 0.235·18-s + 0.335·19-s − 0.774·20-s − 0.213·22-s − 1.44·23-s + 0.204·24-s + 1.39·25-s + 0.392·26-s − 0.192·27-s − 1.11·29-s − 0.632·30-s + 0.262·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 2.92T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 1.07T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 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
−7.980839856472816231512541391835, −7.73075203722081899979590398312, −7.04724458323073659643565916361, −6.12233094968129953908267084798, −5.30632100621679819804951013256, −4.20554096578801022587513917028, −3.68522956008377636478202976477, −2.47323405734273934052503713735, −1.04983271370502453256593629120, 0,
1.04983271370502453256593629120, 2.47323405734273934052503713735, 3.68522956008377636478202976477, 4.20554096578801022587513917028, 5.30632100621679819804951013256, 6.12233094968129953908267084798, 7.04724458323073659643565916361, 7.73075203722081899979590398312, 7.980839856472816231512541391835
