| L(s) = 1 | − 3-s + 5-s − 7-s + 9-s − 11-s + 3.47·13-s − 15-s + 5.23·17-s + 6.70·19-s + 21-s − 5.70·23-s − 4·25-s − 27-s + 5·29-s + 5.23·31-s + 33-s − 35-s − 7·37-s − 3.47·39-s − 2.47·41-s − 5.70·43-s + 45-s − 0.236·47-s + 49-s − 5.23·51-s − 12.1·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.377·7-s + 0.333·9-s − 0.301·11-s + 0.962·13-s − 0.258·15-s + 1.26·17-s + 1.53·19-s + 0.218·21-s − 1.19·23-s − 0.800·25-s − 0.192·27-s + 0.928·29-s + 0.940·31-s + 0.174·33-s − 0.169·35-s − 1.15·37-s − 0.555·39-s − 0.386·41-s − 0.870·43-s + 0.149·45-s − 0.0344·47-s + 0.142·49-s − 0.733·51-s − 1.67·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.710267795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.710267795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 0.236T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 9.76T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 - 4.52T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 + 6.76T + 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 - 9.70T + 97T^{2} \) |
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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
−8.345074731644803461518263267674, −7.888777950753163610755844815408, −6.89419825735184798991796033444, −6.21181566003886440793031680112, −5.57474979918229926290593178050, −4.97525234525818746706608198425, −3.76546895843455998970723201874, −3.14493531295592046753827661780, −1.83170257304283884160587878345, −0.814068508232373021254646778160,
0.814068508232373021254646778160, 1.83170257304283884160587878345, 3.14493531295592046753827661780, 3.76546895843455998970723201874, 4.97525234525818746706608198425, 5.57474979918229926290593178050, 6.21181566003886440793031680112, 6.89419825735184798991796033444, 7.888777950753163610755844815408, 8.345074731644803461518263267674
