| L(s) = 1 | − 3-s + 2.73·5-s − 7-s + 9-s + 6.19·11-s − 2.73·15-s − 4.73·17-s + 0.535·19-s + 21-s + 5.26·23-s + 2.46·25-s − 27-s + 3.46·29-s + 8.92·31-s − 6.19·33-s − 2.73·35-s − 10·37-s − 4.73·41-s + 12.9·43-s + 2.73·45-s − 6.92·47-s + 49-s + 4.73·51-s + 3.46·53-s + 16.9·55-s − 0.535·57-s + 2.92·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.22·5-s − 0.377·7-s + 0.333·9-s + 1.86·11-s − 0.705·15-s − 1.14·17-s + 0.122·19-s + 0.218·21-s + 1.09·23-s + 0.492·25-s − 0.192·27-s + 0.643·29-s + 1.60·31-s − 1.07·33-s − 0.461·35-s − 1.64·37-s − 0.739·41-s + 1.97·43-s + 0.407·45-s − 1.01·47-s + 0.142·49-s + 0.662·51-s + 0.475·53-s + 2.28·55-s − 0.0709·57-s + 0.381·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.316994573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.316994573\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 5 | \( 1 - 2.73T + 5T^{2} \) |
| 11 | \( 1 - 6.19T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.73T + 17T^{2} \) |
| 19 | \( 1 - 0.535T + 19T^{2} \) |
| 23 | \( 1 - 5.26T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 4.73T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 - 3.46T + 53T^{2} \) |
| 59 | \( 1 - 2.92T + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 + 0.535T + 67T^{2} \) |
| 71 | \( 1 - 3.80T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 - 2.53T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 4.73T + 89T^{2} \) |
| 97 | \( 1 + 14.3T + 97T^{2} \) |
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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
−8.418127817264308191447454230740, −6.97043628321764978688023785594, −6.70509081619637486826052883013, −6.17416885115101496306723764960, −5.37308166137076779118576626552, −4.58667583527359396900236784806, −3.80418767014071736899528645652, −2.71992682841692787760051066851, −1.74307990653100790705285033155, −0.897847906683673217973501594445,
0.897847906683673217973501594445, 1.74307990653100790705285033155, 2.71992682841692787760051066851, 3.80418767014071736899528645652, 4.58667583527359396900236784806, 5.37308166137076779118576626552, 6.17416885115101496306723764960, 6.70509081619637486826052883013, 6.97043628321764978688023785594, 8.418127817264308191447454230740
