| L(s) = 1 | + 2.14·2-s + 3-s + 2.59·4-s − 1.47·5-s + 2.14·6-s + 1.27·8-s + 9-s − 3.15·10-s − 2.07·11-s + 2.59·12-s + 2.67·13-s − 1.47·15-s − 2.45·16-s − 4.39·17-s + 2.14·18-s + 0.0316·19-s − 3.82·20-s − 4.43·22-s − 6.77·23-s + 1.27·24-s − 2.83·25-s + 5.73·26-s + 27-s − 4.35·29-s − 3.15·30-s − 0.939·31-s − 7.81·32-s + ⋯ |
| L(s) = 1 | + 1.51·2-s + 0.577·3-s + 1.29·4-s − 0.658·5-s + 0.875·6-s + 0.450·8-s + 0.333·9-s − 0.998·10-s − 0.624·11-s + 0.749·12-s + 0.742·13-s − 0.380·15-s − 0.614·16-s − 1.06·17-s + 0.505·18-s + 0.00725·19-s − 0.854·20-s − 0.946·22-s − 1.41·23-s + 0.260·24-s − 0.566·25-s + 1.12·26-s + 0.192·27-s − 0.808·29-s − 0.576·30-s − 0.168·31-s − 1.38·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 - 2.67T + 13T^{2} \) |
| 17 | \( 1 + 4.39T + 17T^{2} \) |
| 19 | \( 1 - 0.0316T + 19T^{2} \) |
| 23 | \( 1 + 6.77T + 23T^{2} \) |
| 29 | \( 1 + 4.35T + 29T^{2} \) |
| 31 | \( 1 + 0.939T + 31T^{2} \) |
| 37 | \( 1 + 1.46T + 37T^{2} \) |
| 43 | \( 1 + 3.74T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 3.59T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 - 2.01T + 61T^{2} \) |
| 67 | \( 1 + 5.92T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 - 8.42T + 73T^{2} \) |
| 79 | \( 1 - 10.7T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 4.60T + 89T^{2} \) |
| 97 | \( 1 - 3.89T + 97T^{2} \) |
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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
−7.67443620136229960487419114927, −6.86072111148101049919334257282, −6.14802805800256233363274703429, −5.48483011455981603813713408028, −4.62056099409107634837417259183, −3.94757124373235902715408858444, −3.56798962930596426617445218601, −2.58195434332130389142628260904, −1.86553686528441213526094893410, 0,
1.86553686528441213526094893410, 2.58195434332130389142628260904, 3.56798962930596426617445218601, 3.94757124373235902715408858444, 4.62056099409107634837417259183, 5.48483011455981603813713408028, 6.14802805800256233363274703429, 6.86072111148101049919334257282, 7.67443620136229960487419114927
