| L(s) = 1 | − 3-s + 4·7-s + 9-s − 11-s + 4·13-s + 6·17-s + 2·19-s − 4·21-s − 27-s − 4·31-s + 33-s + 10·37-s − 4·39-s + 4·43-s − 12·47-s + 9·49-s − 6·51-s − 6·53-s − 2·57-s + 12·59-s − 10·61-s + 4·63-s + 4·67-s − 8·73-s − 4·77-s − 10·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 1.45·17-s + 0.458·19-s − 0.872·21-s − 0.192·27-s − 0.718·31-s + 0.174·33-s + 1.64·37-s − 0.640·39-s + 0.609·43-s − 1.75·47-s + 9/7·49-s − 0.840·51-s − 0.824·53-s − 0.264·57-s + 1.56·59-s − 1.28·61-s + 0.503·63-s + 0.488·67-s − 0.936·73-s − 0.455·77-s − 1.12·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.113669455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.113669455\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.423552534315639357814459661254, −7.897541068094716892150507054943, −7.32152594890176680420965842688, −6.18645013412678430647532094508, −5.57057844901634901498781537901, −4.92036591675228586311771777183, −4.09579533579079951646323707589, −3.10564868442469072890300856227, −1.73982895103077678067926249332, −0.990938764220696817423527252349,
0.990938764220696817423527252349, 1.73982895103077678067926249332, 3.10564868442469072890300856227, 4.09579533579079951646323707589, 4.92036591675228586311771777183, 5.57057844901634901498781537901, 6.18645013412678430647532094508, 7.32152594890176680420965842688, 7.897541068094716892150507054943, 8.423552534315639357814459661254
