| L(s) = 1 | + 2-s + 3-s + 4-s − 4.41·5-s + 6-s + 8-s + 9-s − 4.41·10-s − 11-s + 12-s − 4.41·15-s + 16-s − 2.37·17-s + 18-s + 3.68·19-s − 4.41·20-s − 22-s + 4.73·23-s + 24-s + 14.5·25-s + 27-s − 6.52·29-s − 4.41·30-s + 3.37·31-s + 32-s − 33-s − 2.37·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.97·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.39·10-s − 0.301·11-s + 0.288·12-s − 1.14·15-s + 0.250·16-s − 0.576·17-s + 0.235·18-s + 0.846·19-s − 0.988·20-s − 0.213·22-s + 0.986·23-s + 0.204·24-s + 2.90·25-s + 0.192·27-s − 1.21·29-s − 0.806·30-s + 0.606·31-s + 0.176·32-s − 0.174·33-s − 0.407·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.401058011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.401058011\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 4.41T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 2.37T + 17T^{2} \) |
| 19 | \( 1 - 3.68T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 + 6.52T + 29T^{2} \) |
| 31 | \( 1 - 3.37T + 31T^{2} \) |
| 37 | \( 1 - 7.68T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 3.06T + 43T^{2} \) |
| 47 | \( 1 - 4.10T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 3.79T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 0.934T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 + 0.418T + 79T^{2} \) |
| 83 | \( 1 - 2.14T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−8.463791525190660396460296221639, −7.79425208276194360671257782929, −7.25576630114647157390665421352, −6.65524345478033424339584669833, −5.32004156754724347588504480485, −4.64076953311351185095331090043, −3.84073990510888471396066896379, −3.33505595028389947601781748158, −2.43085845714481921147854057069, −0.810806301493169226110157436970,
0.810806301493169226110157436970, 2.43085845714481921147854057069, 3.33505595028389947601781748158, 3.84073990510888471396066896379, 4.64076953311351185095331090043, 5.32004156754724347588504480485, 6.65524345478033424339584669833, 7.25576630114647157390665421352, 7.79425208276194360671257782929, 8.463791525190660396460296221639
