| L(s) = 1 | − 3-s + 5-s − 5.23·7-s + 9-s + 2.47·11-s − 13-s − 15-s + 0.763·17-s − 5.23·19-s + 5.23·21-s − 2.76·23-s + 25-s − 27-s + 4.76·29-s + 8.94·31-s − 2.47·33-s − 5.23·35-s − 8.47·37-s + 39-s − 3.52·41-s − 4.94·43-s + 45-s − 12.9·47-s + 20.4·49-s − 0.763·51-s − 8.47·53-s + 2.47·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.97·7-s + 0.333·9-s + 0.745·11-s − 0.277·13-s − 0.258·15-s + 0.185·17-s − 1.20·19-s + 1.14·21-s − 0.576·23-s + 0.200·25-s − 0.192·27-s + 0.884·29-s + 1.60·31-s − 0.430·33-s − 0.885·35-s − 1.39·37-s + 0.160·39-s − 0.550·41-s − 0.753·43-s + 0.149·45-s − 1.88·47-s + 2.91·49-s − 0.106·51-s − 1.16·53-s + 0.333·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9147594837\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9147594837\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 5.23T + 7T^{2} \) |
| 11 | \( 1 - 2.47T + 11T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 19 | \( 1 + 5.23T + 19T^{2} \) |
| 23 | \( 1 + 2.76T + 23T^{2} \) |
| 29 | \( 1 - 4.76T + 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 + 4.94T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 2.47T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 9.70T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.154090016955695965102420801558, −6.79958686774040582709181373527, −6.64522368474562584605209064540, −6.19374884638112717796247916065, −5.29530156640175690830679343184, −4.41737895757354901596239752408, −3.57791525060624502059446009244, −2.87564665754378066604535224666, −1.80672290064474709176169981096, −0.49867092221147740525870604808,
0.49867092221147740525870604808, 1.80672290064474709176169981096, 2.87564665754378066604535224666, 3.57791525060624502059446009244, 4.41737895757354901596239752408, 5.29530156640175690830679343184, 6.19374884638112717796247916065, 6.64522368474562584605209064540, 6.79958686774040582709181373527, 8.154090016955695965102420801558
