L(s) = 1 | + 0.262·2-s + 3-s − 1.93·4-s + 0.855·5-s + 0.262·6-s + 3.34·7-s − 1.03·8-s + 9-s + 0.224·10-s + 5.09·11-s − 1.93·12-s + 13-s + 0.877·14-s + 0.855·15-s + 3.59·16-s − 1.83·17-s + 0.262·18-s + 6.03·19-s − 1.65·20-s + 3.34·21-s + 1.33·22-s + 3.14·23-s − 1.03·24-s − 4.26·25-s + 0.262·26-s + 27-s − 6.46·28-s + ⋯ |
L(s) = 1 | + 0.185·2-s + 0.577·3-s − 0.965·4-s + 0.382·5-s + 0.106·6-s + 1.26·7-s − 0.364·8-s + 0.333·9-s + 0.0708·10-s + 1.53·11-s − 0.557·12-s + 0.277·13-s + 0.234·14-s + 0.220·15-s + 0.898·16-s − 0.444·17-s + 0.0617·18-s + 1.38·19-s − 0.369·20-s + 0.730·21-s + 0.284·22-s + 0.656·23-s − 0.210·24-s − 0.853·25-s + 0.0513·26-s + 0.192·27-s − 1.22·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.097217518\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.097217518\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 0.262T + 2T^{2} \) |
| 5 | \( 1 - 0.855T + 5T^{2} \) |
| 7 | \( 1 - 3.34T + 7T^{2} \) |
| 11 | \( 1 - 5.09T + 11T^{2} \) |
| 17 | \( 1 + 1.83T + 17T^{2} \) |
| 19 | \( 1 - 6.03T + 19T^{2} \) |
| 23 | \( 1 - 3.14T + 23T^{2} \) |
| 29 | \( 1 - 5.61T + 29T^{2} \) |
| 31 | \( 1 - 1.52T + 31T^{2} \) |
| 37 | \( 1 - 0.765T + 37T^{2} \) |
| 41 | \( 1 + 8.75T + 41T^{2} \) |
| 43 | \( 1 + 6.89T + 43T^{2} \) |
| 47 | \( 1 + 7.42T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 3.09T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 6.30T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 6.70T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 8.13T + 89T^{2} \) |
| 97 | \( 1 - 2.06T + 97T^{2} \) |
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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
−8.375025700451917389271846214003, −8.062183823372323429877119551792, −6.97550885989570357920928234207, −6.24083067411281986134917141892, −5.15635691321875762922174387218, −4.74958354529116675365323483894, −3.85648045284024654348392430776, −3.17883391452691077960666104822, −1.77692207639163662168661522213, −1.09223002536061052528357570470,
1.09223002536061052528357570470, 1.77692207639163662168661522213, 3.17883391452691077960666104822, 3.85648045284024654348392430776, 4.74958354529116675365323483894, 5.15635691321875762922174387218, 6.24083067411281986134917141892, 6.97550885989570357920928234207, 8.062183823372323429877119551792, 8.375025700451917389271846214003