| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s + 9-s + 10-s − 2·11-s − 12-s + 13-s + 2·14-s + 15-s + 16-s − 6·17-s − 18-s − 2·19-s − 20-s + 2·21-s + 2·22-s − 6·23-s + 24-s + 25-s − 26-s − 27-s − 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.277·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s + 0.436·21-s + 0.426·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.80745141896105, −16.58510420660524, −15.96321094634312, −15.52419699176287, −15.06387473238426, −14.27361750066927, −13.27890328668400, −13.06423319716768, −12.40496050889003, −11.64493146904525, −11.26690988760127, −10.57587184860667, −10.19256877047578, −9.402744449171254, −8.901296815848353, −8.146778178334215, −7.622256568046246, −6.780087637443507, −6.435993281665436, −5.711125970289542, −4.887448446823873, −4.057018855619631, −3.372757616553378, −2.374857819000470, −1.596887902971062, 0, 0,
1.596887902971062, 2.374857819000470, 3.372757616553378, 4.057018855619631, 4.887448446823873, 5.711125970289542, 6.435993281665436, 6.780087637443507, 7.622256568046246, 8.146778178334215, 8.901296815848353, 9.402744449171254, 10.19256877047578, 10.57587184860667, 11.26690988760127, 11.64493146904525, 12.40496050889003, 13.06423319716768, 13.27890328668400, 14.27361750066927, 15.06387473238426, 15.52419699176287, 15.96321094634312, 16.58510420660524, 16.80745141896105
