| L(s) = 1 | + 3-s − 5-s − 2.59·7-s + 9-s + 0.555·11-s + 5.55·13-s − 15-s − 0.185·17-s − 2.29·19-s − 2.59·21-s − 0.888·23-s + 25-s + 27-s − 1.39·29-s − 3.32·31-s + 0.555·33-s + 2.59·35-s + 7.98·37-s + 5.55·39-s + 5.86·41-s − 11.7·43-s − 45-s + 11.6·47-s − 0.283·49-s − 0.185·51-s + 2.91·53-s − 0.555·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.979·7-s + 0.333·9-s + 0.167·11-s + 1.54·13-s − 0.258·15-s − 0.0450·17-s − 0.525·19-s − 0.565·21-s − 0.185·23-s + 0.200·25-s + 0.192·27-s − 0.259·29-s − 0.596·31-s + 0.0966·33-s + 0.438·35-s + 1.31·37-s + 0.889·39-s + 0.915·41-s − 1.78·43-s − 0.149·45-s + 1.70·47-s − 0.0404·49-s − 0.0260·51-s + 0.401·53-s − 0.0748·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.965311461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.965311461\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 7 | \( 1 + 2.59T + 7T^{2} \) |
| 11 | \( 1 - 0.555T + 11T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 + 0.185T + 17T^{2} \) |
| 19 | \( 1 + 2.29T + 19T^{2} \) |
| 23 | \( 1 + 0.888T + 23T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 + 3.32T + 31T^{2} \) |
| 37 | \( 1 - 7.98T + 37T^{2} \) |
| 41 | \( 1 - 5.86T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 - 2.91T + 53T^{2} \) |
| 59 | \( 1 - 9.00T + 59T^{2} \) |
| 61 | \( 1 - 5.74T + 61T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 3.45T + 73T^{2} \) |
| 79 | \( 1 - 1.73T + 79T^{2} \) |
| 83 | \( 1 - 0.853T + 83T^{2} \) |
| 89 | \( 1 - 14.5T + 89T^{2} \) |
| 97 | \( 1 - 5.48T + 97T^{2} \) |
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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
−8.539569004459527004327639168654, −7.79434066615024616935365536731, −6.97428771297021422279693681843, −6.30791885859013793554707019157, −5.64626042197742304786825606452, −4.36434429129745148811402854217, −3.75256100135535033529430532951, −3.13370909646083734329401925162, −2.06306967510104273748995188417, −0.78194812122568250499648278787,
0.78194812122568250499648278787, 2.06306967510104273748995188417, 3.13370909646083734329401925162, 3.75256100135535033529430532951, 4.36434429129745148811402854217, 5.64626042197742304786825606452, 6.30791885859013793554707019157, 6.97428771297021422279693681843, 7.79434066615024616935365536731, 8.539569004459527004327639168654
