L(s) = 1 | − 2.13·2-s − 2.11·3-s + 2.56·4-s + 3.94·5-s + 4.50·6-s − 1.45·7-s − 1.20·8-s + 1.45·9-s − 8.43·10-s − 11-s − 5.41·12-s − 6.16·13-s + 3.11·14-s − 8.33·15-s − 2.55·16-s + 17-s − 3.10·18-s + 4.81·19-s + 10.1·20-s + 3.07·21-s + 2.13·22-s + 2.82·23-s + 2.54·24-s + 10.5·25-s + 13.1·26-s + 3.26·27-s − 3.73·28-s + ⋯ |
L(s) = 1 | − 1.51·2-s − 1.21·3-s + 1.28·4-s + 1.76·5-s + 1.84·6-s − 0.551·7-s − 0.425·8-s + 0.484·9-s − 2.66·10-s − 0.301·11-s − 1.56·12-s − 1.71·13-s + 0.832·14-s − 2.15·15-s − 0.638·16-s + 0.242·17-s − 0.732·18-s + 1.10·19-s + 2.26·20-s + 0.671·21-s + 0.455·22-s + 0.588·23-s + 0.518·24-s + 2.11·25-s + 2.58·26-s + 0.627·27-s − 0.706·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5601164612\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5601164612\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + 2.13T + 2T^{2} \) |
| 3 | \( 1 + 2.11T + 3T^{2} \) |
| 5 | \( 1 - 3.94T + 5T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 13 | \( 1 + 6.16T + 13T^{2} \) |
| 19 | \( 1 - 4.81T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 1.46T + 29T^{2} \) |
| 31 | \( 1 - 1.97T + 31T^{2} \) |
| 37 | \( 1 + 1.52T + 37T^{2} \) |
| 41 | \( 1 - 8.31T + 41T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 6.70T + 53T^{2} \) |
| 59 | \( 1 - 4.50T + 59T^{2} \) |
| 61 | \( 1 - 8.19T + 61T^{2} \) |
| 67 | \( 1 + 2.49T + 67T^{2} \) |
| 71 | \( 1 + 3.16T + 71T^{2} \) |
| 73 | \( 1 + 3.42T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 2.83T + 83T^{2} \) |
| 89 | \( 1 + 9.91T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−7.73324397781689881908218570480, −7.13145429203708908330118563576, −6.50510515382991225620045543073, −5.96382116655977007439287421201, −5.09636065801720193300485931535, −4.88867417465527244199008810150, −3.01748259827304133826731353647, −2.34140021136697490302615225825, −1.42666211381477004162274911945, −0.52241223855927192679827406179,
0.52241223855927192679827406179, 1.42666211381477004162274911945, 2.34140021136697490302615225825, 3.01748259827304133826731353647, 4.88867417465527244199008810150, 5.09636065801720193300485931535, 5.96382116655977007439287421201, 6.50510515382991225620045543073, 7.13145429203708908330118563576, 7.73324397781689881908218570480