| L(s) = 1 | − 3·3-s + 7·7-s + 9·9-s − 26·11-s − 39·13-s − 101·17-s − 48·19-s − 21·21-s + 57·23-s − 27·27-s − 291·29-s − 79·31-s + 78·33-s + 322·37-s + 117·39-s + 455·41-s − 307·43-s + 508·47-s + 49·49-s + 303·51-s − 31·53-s + 144·57-s + 211·59-s − 797·61-s + 63·63-s − 924·67-s − 171·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.712·11-s − 0.832·13-s − 1.44·17-s − 0.579·19-s − 0.218·21-s + 0.516·23-s − 0.192·27-s − 1.86·29-s − 0.457·31-s + 0.411·33-s + 1.43·37-s + 0.480·39-s + 1.73·41-s − 1.08·43-s + 1.57·47-s + 1/7·49-s + 0.831·51-s − 0.0803·53-s + 0.334·57-s + 0.465·59-s − 1.67·61-s + 0.125·63-s − 1.68·67-s − 0.298·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9430276991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9430276991\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 + 26 T + p^{3} T^{2} \) |
| 13 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 101 T + p^{3} T^{2} \) |
| 19 | \( 1 + 48 T + p^{3} T^{2} \) |
| 23 | \( 1 - 57 T + p^{3} T^{2} \) |
| 29 | \( 1 + 291 T + p^{3} T^{2} \) |
| 31 | \( 1 + 79 T + p^{3} T^{2} \) |
| 37 | \( 1 - 322 T + p^{3} T^{2} \) |
| 41 | \( 1 - 455 T + p^{3} T^{2} \) |
| 43 | \( 1 + 307 T + p^{3} T^{2} \) |
| 47 | \( 1 - 508 T + p^{3} T^{2} \) |
| 53 | \( 1 + 31 T + p^{3} T^{2} \) |
| 59 | \( 1 - 211 T + p^{3} T^{2} \) |
| 61 | \( 1 + 797 T + p^{3} T^{2} \) |
| 67 | \( 1 + 924 T + p^{3} T^{2} \) |
| 71 | \( 1 - 212 T + p^{3} T^{2} \) |
| 73 | \( 1 - 22 T + p^{3} T^{2} \) |
| 79 | \( 1 - 874 T + p^{3} T^{2} \) |
| 83 | \( 1 + 587 T + p^{3} T^{2} \) |
| 89 | \( 1 + 266 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1822 T + p^{3} T^{2} \) |
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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
−8.902290066467846118764020001027, −7.72971932756953662297598701881, −7.32872666951037257100919533520, −6.31617834774423730954499120720, −5.56413411615059705932925016292, −4.72808751150001745091144374003, −4.09507845295283943174457005355, −2.67783289368458826341333594204, −1.88023975921063487974625944762, −0.43802819915040762544266938270,
0.43802819915040762544266938270, 1.88023975921063487974625944762, 2.67783289368458826341333594204, 4.09507845295283943174457005355, 4.72808751150001745091144374003, 5.56413411615059705932925016292, 6.31617834774423730954499120720, 7.32872666951037257100919533520, 7.72971932756953662297598701881, 8.902290066467846118764020001027
