| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 3·7-s + 8-s + 9-s + 10-s + 11-s − 12-s − 13-s − 3·14-s − 15-s + 16-s + 6·17-s + 18-s + 5·19-s + 20-s + 3·21-s + 22-s + 23-s − 24-s + 25-s − 26-s − 27-s − 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 1.14·19-s + 0.223·20-s + 0.654·21-s + 0.213·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.604234831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.604234831\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−12.54082301337108, −12.03512443980154, −11.78979102335180, −11.24320849961227, −10.71163735834424, −10.15533806838627, −9.865148703549362, −9.498387280749151, −9.093276088788698, −8.206939015026144, −7.831352159754965, −7.262414067827663, −6.797636682641059, −6.352320423465187, −6.032967456510767, −5.429557787126507, −5.145127425325265, −4.585016110451003, −3.934719362370970, −3.328441637032249, −3.079933046918081, −2.488721004052222, −1.692882158127759, −1.037638867416904, −0.5769222978769323,
0.5769222978769323, 1.037638867416904, 1.692882158127759, 2.488721004052222, 3.079933046918081, 3.328441637032249, 3.934719362370970, 4.585016110451003, 5.145127425325265, 5.429557787126507, 6.032967456510767, 6.352320423465187, 6.797636682641059, 7.262414067827663, 7.831352159754965, 8.206939015026144, 9.093276088788698, 9.498387280749151, 9.865148703549362, 10.15533806838627, 10.71163735834424, 11.24320849961227, 11.78979102335180, 12.03512443980154, 12.54082301337108
