| L(s) = 1 | − 1.61·2-s − 3-s + 0.618·4-s + 0.618·5-s + 1.61·6-s − 7-s + 2.23·8-s + 9-s − 1.00·10-s − 0.618·12-s − 0.236·13-s + 1.61·14-s − 0.618·15-s − 4.85·16-s + 5.47·17-s − 1.61·18-s − 2.47·19-s + 0.381·20-s + 21-s − 7.70·23-s − 2.23·24-s − 4.61·25-s + 0.381·26-s − 27-s − 0.618·28-s + 4.23·29-s + 1.00·30-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 0.577·3-s + 0.309·4-s + 0.276·5-s + 0.660·6-s − 0.377·7-s + 0.790·8-s + 0.333·9-s − 0.316·10-s − 0.178·12-s − 0.0654·13-s + 0.432·14-s − 0.159·15-s − 1.21·16-s + 1.32·17-s − 0.381·18-s − 0.567·19-s + 0.0854·20-s + 0.218·21-s − 1.60·23-s − 0.456·24-s − 0.923·25-s + 0.0749·26-s − 0.192·27-s − 0.116·28-s + 0.786·29-s + 0.182·30-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 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 - 0.618T + 5T^{2} \) |
| 13 | \( 1 + 0.236T + 13T^{2} \) |
| 17 | \( 1 - 5.47T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 + 7.70T + 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 - 3T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 3.14T + 41T^{2} \) |
| 43 | \( 1 + 2.52T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 7.56T + 53T^{2} \) |
| 59 | \( 1 + 2.38T + 59T^{2} \) |
| 61 | \( 1 - 9.94T + 61T^{2} \) |
| 67 | \( 1 - 8.76T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 - 9.61T + 73T^{2} \) |
| 79 | \( 1 - 5.09T + 79T^{2} \) |
| 83 | \( 1 + 0.472T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + 18.7T + 97T^{2} \) |
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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
−8.327934376947044845302522964886, −8.074059882382559927337142399559, −7.09357399726534495043827880342, −6.30404395018599811616369310528, −5.57471051272296889306402188530, −4.58080195478926575436748337469, −3.68605937965413176534352486313, −2.27706129421319637198052010592, −1.20790170200073094362618786995, 0,
1.20790170200073094362618786995, 2.27706129421319637198052010592, 3.68605937965413176534352486313, 4.58080195478926575436748337469, 5.57471051272296889306402188530, 6.30404395018599811616369310528, 7.09357399726534495043827880342, 8.074059882382559927337142399559, 8.327934376947044845302522964886
