| L(s) = 1 | + 2.71·2-s + 5.37·4-s + 3.22·5-s + 9.15·8-s + 8.74·10-s − 2.20·11-s − 2·13-s + 14.1·16-s + 3.22·17-s − 19-s + 17.3·20-s − 5.99·22-s + 1.01·23-s + 5.37·25-s − 5.43·26-s + 1.01·29-s − 4.74·31-s + 20.0·32-s + 8.74·34-s + 10.7·37-s − 2.71·38-s + 29.4·40-s + 5.43·41-s − 11.1·43-s − 11.8·44-s + 2.74·46-s + 4.23·47-s + ⋯ |
| L(s) = 1 | + 1.91·2-s + 2.68·4-s + 1.44·5-s + 3.23·8-s + 2.76·10-s − 0.666·11-s − 0.554·13-s + 3.52·16-s + 0.781·17-s − 0.229·19-s + 3.86·20-s − 1.27·22-s + 0.210·23-s + 1.07·25-s − 1.06·26-s + 0.187·29-s − 0.852·31-s + 3.53·32-s + 1.49·34-s + 1.76·37-s − 0.440·38-s + 4.66·40-s + 0.848·41-s − 1.69·43-s − 1.78·44-s + 0.404·46-s + 0.617·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.38883191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.38883191\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.71T + 2T^{2} \) |
| 5 | \( 1 - 3.22T + 5T^{2} \) |
| 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 3.22T + 17T^{2} \) |
| 23 | \( 1 - 1.01T + 23T^{2} \) |
| 29 | \( 1 - 1.01T + 29T^{2} \) |
| 31 | \( 1 + 4.74T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 4.23T + 47T^{2} \) |
| 53 | \( 1 - 9.84T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 - 5.11T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 2.02T + 71T^{2} \) |
| 73 | \( 1 - 5.11T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 11.8T + 83T^{2} \) |
| 89 | \( 1 + 9.84T + 89T^{2} \) |
| 97 | \( 1 + 7.48T + 97T^{2} \) |
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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
−7.44757110507443960519736963386, −6.81034540783600481844910416464, −6.05847974417825247414370593848, −5.63923072551832079692383085755, −5.10685040813553007781398668570, −4.44645349339399211178669762350, −3.51925593110544982509969057103, −2.66503392485324049057957048954, −2.25953573699197105633259428492, −1.31067211067296379370270793056,
1.31067211067296379370270793056, 2.25953573699197105633259428492, 2.66503392485324049057957048954, 3.51925593110544982509969057103, 4.44645349339399211178669762350, 5.10685040813553007781398668570, 5.63923072551832079692383085755, 6.05847974417825247414370593848, 6.81034540783600481844910416464, 7.44757110507443960519736963386
