| L(s) = 1 | − 2-s + 3-s + 4-s − 0.390·5-s − 6-s − 3.35·7-s − 8-s + 9-s + 0.390·10-s + 2.35·11-s + 12-s + 0.685·13-s + 3.35·14-s − 0.390·15-s + 16-s − 17-s − 18-s − 3.67·19-s − 0.390·20-s − 3.35·21-s − 2.35·22-s + 0.781·23-s − 24-s − 4.84·25-s − 0.685·26-s + 27-s − 3.35·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.174·5-s − 0.408·6-s − 1.26·7-s − 0.353·8-s + 0.333·9-s + 0.123·10-s + 0.710·11-s + 0.288·12-s + 0.189·13-s + 0.897·14-s − 0.100·15-s + 0.250·16-s − 0.242·17-s − 0.235·18-s − 0.843·19-s − 0.0873·20-s − 0.732·21-s − 0.502·22-s + 0.163·23-s − 0.204·24-s − 0.969·25-s − 0.134·26-s + 0.192·27-s − 0.634·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 + 0.390T + 5T^{2} \) |
| 7 | \( 1 + 3.35T + 7T^{2} \) |
| 11 | \( 1 - 2.35T + 11T^{2} \) |
| 13 | \( 1 - 0.685T + 13T^{2} \) |
| 19 | \( 1 + 3.67T + 19T^{2} \) |
| 23 | \( 1 - 0.781T + 23T^{2} \) |
| 29 | \( 1 - 8.31T + 29T^{2} \) |
| 31 | \( 1 - 1.31T + 31T^{2} \) |
| 37 | \( 1 - 5.37T + 37T^{2} \) |
| 41 | \( 1 - 8.37T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 + 6.66T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 61 | \( 1 - 9.75T + 61T^{2} \) |
| 67 | \( 1 - 2.59T + 67T^{2} \) |
| 71 | \( 1 + 8.47T + 71T^{2} \) |
| 73 | \( 1 + 1.74T + 73T^{2} \) |
| 79 | \( 1 - 6.53T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 7.04T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.983941002442667423380581010835, −6.92220132285647754728363987019, −6.53328842172100177714668541240, −5.95567630557781758214079667645, −4.63947728690956739438433467366, −3.85330245925910597745750149273, −3.11361700900922028560069110658, −2.36028458392840841973011882398, −1.24889333455046511973041115310, 0,
1.24889333455046511973041115310, 2.36028458392840841973011882398, 3.11361700900922028560069110658, 3.85330245925910597745750149273, 4.63947728690956739438433467366, 5.95567630557781758214079667645, 6.53328842172100177714668541240, 6.92220132285647754728363987019, 7.983941002442667423380581010835
