| L(s) = 1 | − 3-s − 2·4-s − 5-s + 7-s + 9-s + 4·11-s + 2·12-s + 15-s + 4·16-s + 5·17-s + 2·20-s − 21-s + 23-s + 25-s − 27-s − 2·28-s + 5·29-s + 3·31-s − 4·33-s − 35-s − 2·36-s − 5·37-s + 3·41-s − 4·43-s − 8·44-s − 45-s + 6·47-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.577·12-s + 0.258·15-s + 16-s + 1.21·17-s + 0.447·20-s − 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.377·28-s + 0.928·29-s + 0.538·31-s − 0.696·33-s − 0.169·35-s − 1/3·36-s − 0.821·37-s + 0.468·41-s − 0.609·43-s − 1.20·44-s − 0.149·45-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9180584301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9180584301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−11.75563198503866117977232289242, −10.54220297670747207171532732965, −9.686930432979919423719343062050, −8.721946943978868929572098933577, −7.84363943487410374278537021794, −6.65192775019186314723270629428, −5.46143282343266684921383536963, −4.51296358389313921281668173062, −3.53278255543219487448641156845, −1.06926450351732859500707528805,
1.06926450351732859500707528805, 3.53278255543219487448641156845, 4.51296358389313921281668173062, 5.46143282343266684921383536963, 6.65192775019186314723270629428, 7.84363943487410374278537021794, 8.721946943978868929572098933577, 9.686930432979919423719343062050, 10.54220297670747207171532732965, 11.75563198503866117977232289242
