| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s + 2.82·7-s − 3·8-s + 9-s + 0.828·11-s + 12-s + 5.65·13-s + 2.82·14-s − 16-s − 3.65·17-s + 18-s + 0.828·19-s − 2.82·21-s + 0.828·22-s − 8.48·23-s + 3·24-s + 5.65·26-s − 27-s − 2.82·28-s − 29-s + 4.82·31-s + 5·32-s − 0.828·33-s − 3.65·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.5·4-s − 0.408·6-s + 1.06·7-s − 1.06·8-s + 0.333·9-s + 0.249·11-s + 0.288·12-s + 1.56·13-s + 0.755·14-s − 0.250·16-s − 0.886·17-s + 0.235·18-s + 0.190·19-s − 0.617·21-s + 0.176·22-s − 1.76·23-s + 0.612·24-s + 1.10·26-s − 0.192·27-s − 0.534·28-s − 0.185·29-s + 0.867·31-s + 0.883·32-s − 0.144·33-s − 0.627·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.966486101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.966486101\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 3.65T + 17T^{2} \) |
| 19 | \( 1 - 0.828T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 31 | \( 1 - 4.82T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 6.82T + 67T^{2} \) |
| 71 | \( 1 + 1.65T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 + 8.82T + 79T^{2} \) |
| 83 | \( 1 - 12.4T + 83T^{2} \) |
| 89 | \( 1 + 0.343T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.883371750441802877587363890679, −8.405296979257659138142115677797, −7.54259416172157424817093696745, −6.21679913994675243936255238605, −5.99295448865948099082386597998, −4.96213733696130331899894350097, −4.26171125290291352201843503681, −3.69208276688899387848246405969, −2.18616737600283475906594422304, −0.886981989811700912002777893135,
0.886981989811700912002777893135, 2.18616737600283475906594422304, 3.69208276688899387848246405969, 4.26171125290291352201843503681, 4.96213733696130331899894350097, 5.99295448865948099082386597998, 6.21679913994675243936255238605, 7.54259416172157424817093696745, 8.405296979257659138142115677797, 8.883371750441802877587363890679
