| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s + 5.21·11-s + 12-s + 0.789·13-s + 2·14-s + 16-s − 0.115·17-s + 18-s + 0.115·19-s + 2·21-s + 5.21·22-s − 4.42·23-s + 24-s + 0.789·26-s + 27-s + 2·28-s + 4.42·29-s − 31-s + 32-s + 5.21·33-s − 0.115·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 0.333·9-s + 1.57·11-s + 0.288·12-s + 0.219·13-s + 0.534·14-s + 0.250·16-s − 0.0279·17-s + 0.235·18-s + 0.0264·19-s + 0.436·21-s + 1.11·22-s − 0.921·23-s + 0.204·24-s + 0.154·26-s + 0.192·27-s + 0.377·28-s + 0.820·29-s − 0.179·31-s + 0.176·32-s + 0.906·33-s − 0.0197·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.904674102\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.904674102\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 5.21T + 11T^{2} \) |
| 13 | \( 1 - 0.789T + 13T^{2} \) |
| 17 | \( 1 + 0.115T + 17T^{2} \) |
| 19 | \( 1 - 0.115T + 19T^{2} \) |
| 23 | \( 1 + 4.42T + 23T^{2} \) |
| 29 | \( 1 - 4.42T + 29T^{2} \) |
| 37 | \( 1 - 6.61T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.61T + 43T^{2} \) |
| 47 | \( 1 - 0.115T + 47T^{2} \) |
| 53 | \( 1 + 0.190T + 53T^{2} \) |
| 59 | \( 1 + 4.19T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 - 5.82T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 2.30T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 2.23T + 89T^{2} \) |
| 97 | \( 1 + 9.21T + 97T^{2} \) |
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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
−8.253742826083962301616952991508, −7.61923212852350954522068127618, −6.70183141039366571111400964838, −6.23352035568258555413799655221, −5.25684490067060701954155744952, −4.37098813119203208882724021745, −3.92755062400044117387648243703, −3.01722486010701818453434503045, −1.97643656955873048076356377687, −1.21388883770716023775771094286,
1.21388883770716023775771094286, 1.97643656955873048076356377687, 3.01722486010701818453434503045, 3.92755062400044117387648243703, 4.37098813119203208882724021745, 5.25684490067060701954155744952, 6.23352035568258555413799655221, 6.70183141039366571111400964838, 7.61923212852350954522068127618, 8.253742826083962301616952991508
