| L(s) = 1 | − 2-s − 3-s + 4-s − 2·5-s + 6-s − 8-s + 9-s + 2·10-s − 4·11-s − 12-s − 13-s + 2·15-s + 16-s − 6·17-s − 18-s + 4·19-s − 2·20-s + 4·22-s + 24-s − 25-s + 26-s − 27-s − 4·29-s − 2·30-s − 4·31-s − 32-s + 4·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.632·10-s − 1.20·11-s − 0.288·12-s − 0.277·13-s + 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.852·22-s + 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s − 0.742·29-s − 0.365·30-s − 0.718·31-s − 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3277196049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3277196049\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.344766152065386763738727924888, −7.76301146123520548010803974919, −7.20597972637727480355522782024, −6.45758111690053083222291796761, −5.49229591607937282142761674075, −4.81391686765430462961978642005, −3.87181370203354337672917792856, −2.86640654393961297051111689636, −1.84066423816495873856718276447, −0.36344521169304185205797808682,
0.36344521169304185205797808682, 1.84066423816495873856718276447, 2.86640654393961297051111689636, 3.87181370203354337672917792856, 4.81391686765430462961978642005, 5.49229591607937282142761674075, 6.45758111690053083222291796761, 7.20597972637727480355522782024, 7.76301146123520548010803974919, 8.344766152065386763738727924888
