L(s) = 1 | − 2·3-s − 2·4-s − 3·5-s − 7-s + 9-s − 3·11-s + 4·12-s + 4·13-s + 6·15-s + 4·16-s − 3·17-s − 19-s + 6·20-s + 2·21-s + 4·25-s + 4·27-s + 2·28-s − 6·29-s + 4·31-s + 6·33-s + 3·35-s − 2·36-s + 2·37-s − 8·39-s + 6·41-s + 43-s + 6·44-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 4-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.15·12-s + 1.10·13-s + 1.54·15-s + 16-s − 0.727·17-s − 0.229·19-s + 1.34·20-s + 0.436·21-s + 4/5·25-s + 0.769·27-s + 0.377·28-s − 1.11·29-s + 0.718·31-s + 1.04·33-s + 0.507·35-s − 1/3·36-s + 0.328·37-s − 1.28·39-s + 0.937·41-s + 0.152·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41971 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41971 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6019982966\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6019982966\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 + T \) |
| 47 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.82062993720606, −14.17554838271630, −13.50924567493881, −13.03471210825680, −12.72711230000074, −12.07151758943378, −11.59054147429658, −11.08182005207214, −10.69725351046066, −10.20465328287405, −9.376933256368812, −8.872329961180960, −8.300762533956756, −7.840381302542135, −7.305031007172905, −6.411634217398548, −6.066246078508152, −5.369590393183792, −4.805546356798797, −4.314432567977842, −3.679847158564805, −3.208837694698318, −2.132348298031530, −0.7432226443089781, −0.4915439028913777,
0.4915439028913777, 0.7432226443089781, 2.132348298031530, 3.208837694698318, 3.679847158564805, 4.314432567977842, 4.805546356798797, 5.369590393183792, 6.066246078508152, 6.411634217398548, 7.305031007172905, 7.840381302542135, 8.300762533956756, 8.872329961180960, 9.376933256368812, 10.20465328287405, 10.69725351046066, 11.08182005207214, 11.59054147429658, 12.07151758943378, 12.72711230000074, 13.03471210825680, 13.50924567493881, 14.17554838271630, 14.82062993720606