| L(s) = 1 | + 3-s − 0.430·5-s − 7-s + 9-s + 4.67·11-s − 4.61·13-s − 0.430·15-s + 5.52·17-s + 19-s − 21-s + 1.98·23-s − 4.81·25-s + 27-s + 6.41·29-s + 6.95·31-s + 4.67·33-s + 0.430·35-s − 3.95·37-s − 4.61·39-s − 0.487·41-s + 2.25·43-s − 0.430·45-s − 7.10·47-s + 49-s + 5.52·51-s − 7.50·53-s − 2.01·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.192·5-s − 0.377·7-s + 0.333·9-s + 1.40·11-s − 1.27·13-s − 0.111·15-s + 1.33·17-s + 0.229·19-s − 0.218·21-s + 0.413·23-s − 0.962·25-s + 0.192·27-s + 1.19·29-s + 1.24·31-s + 0.813·33-s + 0.0728·35-s − 0.650·37-s − 0.738·39-s − 0.0760·41-s + 0.343·43-s − 0.0642·45-s − 1.03·47-s + 0.142·49-s + 0.773·51-s − 1.03·53-s − 0.271·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.527769283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.527769283\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 0.430T + 5T^{2} \) |
| 11 | \( 1 - 4.67T + 11T^{2} \) |
| 13 | \( 1 + 4.61T + 13T^{2} \) |
| 17 | \( 1 - 5.52T + 17T^{2} \) |
| 23 | \( 1 - 1.98T + 23T^{2} \) |
| 29 | \( 1 - 6.41T + 29T^{2} \) |
| 31 | \( 1 - 6.95T + 31T^{2} \) |
| 37 | \( 1 + 3.95T + 37T^{2} \) |
| 41 | \( 1 + 0.487T + 41T^{2} \) |
| 43 | \( 1 - 2.25T + 43T^{2} \) |
| 47 | \( 1 + 7.10T + 47T^{2} \) |
| 53 | \( 1 + 7.50T + 53T^{2} \) |
| 59 | \( 1 - 2.68T + 59T^{2} \) |
| 61 | \( 1 + 2.22T + 61T^{2} \) |
| 67 | \( 1 + 0.358T + 67T^{2} \) |
| 71 | \( 1 - 13.0T + 71T^{2} \) |
| 73 | \( 1 + 1.77T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 2.40T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 3.22T + 97T^{2} \) |
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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
−8.021421553131908351479637221220, −7.35894221813761787583569205939, −6.71061476173360549873550935617, −6.04302837327401264672559362630, −5.02779122166205476978502076060, −4.36409991301312321303750149613, −3.47010489972900757088139816230, −2.93760432606880505512318181173, −1.84891095555823572812632389992, −0.824627860080261479350754332583,
0.824627860080261479350754332583, 1.84891095555823572812632389992, 2.93760432606880505512318181173, 3.47010489972900757088139816230, 4.36409991301312321303750149613, 5.02779122166205476978502076060, 6.04302837327401264672559362630, 6.71061476173360549873550935617, 7.35894221813761787583569205939, 8.021421553131908351479637221220
