| L(s) = 1 | − 2.52·2-s − 3-s + 4.39·4-s + 0.133·5-s + 2.52·6-s + 7-s − 6.05·8-s + 9-s − 0.337·10-s − 4.39·12-s − 0.133·13-s − 2.52·14-s − 0.133·15-s + 6.52·16-s + 5.05·17-s − 2.52·18-s + 0.924·19-s + 0.586·20-s − 21-s − 7.05·23-s + 6.05·24-s − 4.98·25-s + 0.337·26-s − 27-s + 4.39·28-s − 3.86·29-s + 0.337·30-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 0.577·3-s + 2.19·4-s + 0.0596·5-s + 1.03·6-s + 0.377·7-s − 2.14·8-s + 0.333·9-s − 0.106·10-s − 1.26·12-s − 0.0370·13-s − 0.675·14-s − 0.0344·15-s + 1.63·16-s + 1.22·17-s − 0.596·18-s + 0.212·19-s + 0.131·20-s − 0.218·21-s − 1.47·23-s + 1.23·24-s − 0.996·25-s + 0.0662·26-s − 0.192·27-s + 0.830·28-s − 0.717·29-s + 0.0616·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.52T + 2T^{2} \) |
| 5 | \( 1 - 0.133T + 5T^{2} \) |
| 13 | \( 1 + 0.133T + 13T^{2} \) |
| 17 | \( 1 - 5.05T + 17T^{2} \) |
| 19 | \( 1 - 0.924T + 19T^{2} \) |
| 23 | \( 1 + 7.05T + 23T^{2} \) |
| 29 | \( 1 + 3.86T + 29T^{2} \) |
| 31 | \( 1 - 2.79T + 31T^{2} \) |
| 37 | \( 1 - 9.98T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 3.05T + 43T^{2} \) |
| 47 | \( 1 + 3.07T + 47T^{2} \) |
| 53 | \( 1 + 4.79T + 53T^{2} \) |
| 59 | \( 1 + 12.6T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 8.92T + 67T^{2} \) |
| 71 | \( 1 + 6.11T + 71T^{2} \) |
| 73 | \( 1 + 7.86T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 1.20T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.471862104579938758461367549794, −7.74390791486910267707737602891, −7.46440575658432064381978622417, −6.25474479540350008195764129505, −5.85635901327649688471661303964, −4.64472799360655201724119852120, −3.37787895673199403272734170482, −2.09188453288757583930424761712, −1.26689683958678299750372743870, 0,
1.26689683958678299750372743870, 2.09188453288757583930424761712, 3.37787895673199403272734170482, 4.64472799360655201724119852120, 5.85635901327649688471661303964, 6.25474479540350008195764129505, 7.46440575658432064381978622417, 7.74390791486910267707737602891, 8.471862104579938758461367549794
