| L(s) = 1 | + 3-s + 7-s + 9-s − 3.77·11-s − 2.77·13-s + 4.77·17-s + 21-s + 3·23-s + 27-s + 3·29-s + 4.77·31-s − 3.77·33-s − 7.77·37-s − 2.77·39-s − 2.77·41-s + 12.5·43-s + 49-s + 4.77·51-s + 4.77·53-s − 2.77·59-s + 6.77·61-s + 63-s − 9.77·67-s + 3·69-s + 9.77·71-s + 2·73-s − 3.77·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 0.333·9-s − 1.13·11-s − 0.768·13-s + 1.15·17-s + 0.218·21-s + 0.625·23-s + 0.192·27-s + 0.557·29-s + 0.857·31-s − 0.656·33-s − 1.27·37-s − 0.443·39-s − 0.432·41-s + 1.91·43-s + 0.142·49-s + 0.668·51-s + 0.655·53-s − 0.360·59-s + 0.867·61-s + 0.125·63-s − 1.19·67-s + 0.361·69-s + 1.15·71-s + 0.234·73-s − 0.429·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.361639012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.361639012\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + 3.77T + 11T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 - 4.77T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 3T + 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 4.77T + 31T^{2} \) |
| 37 | \( 1 + 7.77T + 37T^{2} \) |
| 41 | \( 1 + 2.77T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 4.77T + 53T^{2} \) |
| 59 | \( 1 + 2.77T + 59T^{2} \) |
| 61 | \( 1 - 6.77T + 61T^{2} \) |
| 67 | \( 1 + 9.77T + 67T^{2} \) |
| 71 | \( 1 - 9.77T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 - 7.22T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.332415593367354649412188304588, −7.66154539620891145705073920193, −7.25341148855421734311273443624, −6.20345283025833721218893465233, −5.21684064892742000866152506469, −4.82239798288149268810433593492, −3.69405090843577705894061783077, −2.85467461114112516815591297548, −2.15041287886868026498722754617, −0.851612214917272711163293436972,
0.851612214917272711163293436972, 2.15041287886868026498722754617, 2.85467461114112516815591297548, 3.69405090843577705894061783077, 4.82239798288149268810433593492, 5.21684064892742000866152506469, 6.20345283025833721218893465233, 7.25341148855421734311273443624, 7.66154539620891145705073920193, 8.332415593367354649412188304588
