| L(s) = 1 | + 2.41·3-s − 0.704·5-s + 2.82·9-s + 0.778·11-s − 2.49·13-s − 1.69·15-s + 0.435·17-s − 6.91·19-s + 23-s − 4.50·25-s − 0.433·27-s + 6.61·29-s − 2.81·31-s + 1.87·33-s − 2.12·37-s − 6.01·39-s + 7.77·41-s + 2.78·43-s − 1.98·45-s − 8.86·47-s + 1.04·51-s + 5.77·53-s − 0.548·55-s − 16.6·57-s − 3.68·59-s − 3.41·61-s + 1.75·65-s + ⋯ |
| L(s) = 1 | + 1.39·3-s − 0.314·5-s + 0.940·9-s + 0.234·11-s − 0.691·13-s − 0.438·15-s + 0.105·17-s − 1.58·19-s + 0.208·23-s − 0.900·25-s − 0.0834·27-s + 1.22·29-s − 0.505·31-s + 0.326·33-s − 0.349·37-s − 0.963·39-s + 1.21·41-s + 0.424·43-s − 0.295·45-s − 1.29·47-s + 0.147·51-s + 0.793·53-s − 0.0738·55-s − 2.21·57-s − 0.479·59-s − 0.436·61-s + 0.217·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 + 0.704T + 5T^{2} \) |
| 11 | \( 1 - 0.778T + 11T^{2} \) |
| 13 | \( 1 + 2.49T + 13T^{2} \) |
| 17 | \( 1 - 0.435T + 17T^{2} \) |
| 19 | \( 1 + 6.91T + 19T^{2} \) |
| 29 | \( 1 - 6.61T + 29T^{2} \) |
| 31 | \( 1 + 2.81T + 31T^{2} \) |
| 37 | \( 1 + 2.12T + 37T^{2} \) |
| 41 | \( 1 - 7.77T + 41T^{2} \) |
| 43 | \( 1 - 2.78T + 43T^{2} \) |
| 47 | \( 1 + 8.86T + 47T^{2} \) |
| 53 | \( 1 - 5.77T + 53T^{2} \) |
| 59 | \( 1 + 3.68T + 59T^{2} \) |
| 61 | \( 1 + 3.41T + 61T^{2} \) |
| 67 | \( 1 + 5.70T + 67T^{2} \) |
| 71 | \( 1 - 6.33T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 - 1.16T + 79T^{2} \) |
| 83 | \( 1 - 3.12T + 83T^{2} \) |
| 89 | \( 1 - 9.70T + 89T^{2} \) |
| 97 | \( 1 - 3.14T + 97T^{2} \) |
| 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
−7.67320337129202449429541244770, −6.83712041524622147393112322487, −6.19234994058563514291525598366, −5.20405618844654410220840645084, −4.27225380351952788497017803646, −3.89753863649895628485185649527, −2.90950762991870911216148839237, −2.39405579254714589325422791969, −1.51425067217497288042601299131, 0,
1.51425067217497288042601299131, 2.39405579254714589325422791969, 2.90950762991870911216148839237, 3.89753863649895628485185649527, 4.27225380351952788497017803646, 5.20405618844654410220840645084, 6.19234994058563514291525598366, 6.83712041524622147393112322487, 7.67320337129202449429541244770
