| L(s) = 1 | + 0.414·3-s + 5-s − 2.82·9-s + 0.828·11-s + 2·13-s + 0.414·15-s − 7.65·17-s − 5.65·19-s − 5.58·23-s + 25-s − 2.41·27-s − 7.82·29-s + 0.828·31-s + 0.343·33-s + 5.65·37-s + 0.828·39-s + 5.82·41-s − 6.89·43-s − 2.82·45-s + 11.6·47-s − 3.17·51-s − 5.65·53-s + 0.828·55-s − 2.34·57-s − 4·59-s + 6.65·61-s + 2·65-s + ⋯ |
| L(s) = 1 | + 0.239·3-s + 0.447·5-s − 0.942·9-s + 0.249·11-s + 0.554·13-s + 0.106·15-s − 1.85·17-s − 1.29·19-s − 1.16·23-s + 0.200·25-s − 0.464·27-s − 1.45·29-s + 0.148·31-s + 0.0597·33-s + 0.929·37-s + 0.132·39-s + 0.910·41-s − 1.05·43-s − 0.421·45-s + 1.70·47-s − 0.444·51-s − 0.777·53-s + 0.111·55-s − 0.310·57-s − 0.520·59-s + 0.852·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 7.65T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 5.58T + 23T^{2} \) |
| 29 | \( 1 + 7.82T + 29T^{2} \) |
| 31 | \( 1 - 0.828T + 31T^{2} \) |
| 37 | \( 1 - 5.65T + 37T^{2} \) |
| 41 | \( 1 - 5.82T + 41T^{2} \) |
| 43 | \( 1 + 6.89T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6.65T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 - 3.65T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 4.75T + 83T^{2} \) |
| 89 | \( 1 + 5.34T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.834211035385820467997563880257, −8.220199045066586983224569626822, −7.20311365171144219768442050524, −6.14309172842128982033781547984, −5.92658564617297668241953228398, −4.56668904680839968782865697046, −3.86269375915728988627128897120, −2.60316358748980387249469785974, −1.87334374716229158862605822539, 0,
1.87334374716229158862605822539, 2.60316358748980387249469785974, 3.86269375915728988627128897120, 4.56668904680839968782865697046, 5.92658564617297668241953228398, 6.14309172842128982033781547984, 7.20311365171144219768442050524, 8.220199045066586983224569626822, 8.834211035385820467997563880257
