| L(s) = 1 | − 3-s − 3.42·7-s + 9-s + 5.73·11-s + 0.116·13-s − 7.04·17-s + 1.31·19-s + 3.42·21-s + 1.92·23-s − 27-s + 2.85·29-s − 7.89·31-s − 5.73·33-s − 0.813·37-s − 0.116·39-s − 6.66·41-s + 5.70·43-s + 9.84·47-s + 4.73·49-s + 7.04·51-s + 4.39·53-s − 1.31·57-s + 5.04·59-s + 0.970·61-s − 3.42·63-s − 6.09·67-s − 1.92·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.29·7-s + 0.333·9-s + 1.72·11-s + 0.0323·13-s − 1.70·17-s + 0.301·19-s + 0.747·21-s + 0.401·23-s − 0.192·27-s + 0.529·29-s − 1.41·31-s − 0.997·33-s − 0.133·37-s − 0.0186·39-s − 1.04·41-s + 0.870·43-s + 1.43·47-s + 0.675·49-s + 0.985·51-s + 0.603·53-s − 0.173·57-s + 0.657·59-s + 0.124·61-s − 0.431·63-s − 0.744·67-s − 0.231·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.42T + 7T^{2} \) |
| 11 | \( 1 - 5.73T + 11T^{2} \) |
| 13 | \( 1 - 0.116T + 13T^{2} \) |
| 17 | \( 1 + 7.04T + 17T^{2} \) |
| 19 | \( 1 - 1.31T + 19T^{2} \) |
| 23 | \( 1 - 1.92T + 23T^{2} \) |
| 29 | \( 1 - 2.85T + 29T^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 + 0.813T + 37T^{2} \) |
| 41 | \( 1 + 6.66T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 - 9.84T + 47T^{2} \) |
| 53 | \( 1 - 4.39T + 53T^{2} \) |
| 59 | \( 1 - 5.04T + 59T^{2} \) |
| 61 | \( 1 - 0.970T + 61T^{2} \) |
| 67 | \( 1 + 6.09T + 67T^{2} \) |
| 71 | \( 1 + 6.08T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 4.73T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 7.37T + 97T^{2} \) |
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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
−8.650665809785014649392663335105, −7.20305138256450010479777782769, −6.78863374139174621766650647223, −6.23969907239242197255971816402, −5.41502603674012578062355412580, −4.24865643906729250313568657947, −3.76721326222479689109317461427, −2.62394879733219069902123799911, −1.34142434991149659103037998140, 0,
1.34142434991149659103037998140, 2.62394879733219069902123799911, 3.76721326222479689109317461427, 4.24865643906729250313568657947, 5.41502603674012578062355412580, 6.23969907239242197255971816402, 6.78863374139174621766650647223, 7.20305138256450010479777782769, 8.650665809785014649392663335105
