| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 5·5-s + 6·6-s − 1.42·7-s + 8·8-s + 9·9-s + 10·10-s − 25.0·11-s + 12·12-s + 47.9·13-s − 2.85·14-s + 15·15-s + 16·16-s + 77.5·17-s + 18·18-s + 19·19-s + 20·20-s − 4.27·21-s − 50.0·22-s − 53.1·23-s + 24·24-s + 25·25-s + 95.8·26-s + 27·27-s − 5.70·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.0770·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.686·11-s + 0.288·12-s + 1.02·13-s − 0.0544·14-s + 0.258·15-s + 0.250·16-s + 1.10·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s − 0.0444·21-s − 0.485·22-s − 0.482·23-s + 0.204·24-s + 0.200·25-s + 0.722·26-s + 0.192·27-s − 0.0385·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.446795559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.446795559\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 19 | \( 1 - 19T \) |
| good | 7 | \( 1 + 1.42T + 343T^{2} \) |
| 11 | \( 1 + 25.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 47.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 77.5T + 4.91e3T^{2} \) |
| 23 | \( 1 + 53.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 56.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 227.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 372.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 154.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 619.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 285.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 121.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 58.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 74.7T + 3.00e5T^{2} \) |
| 71 | \( 1 + 267.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 144.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 368.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 208.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 944.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 326.T + 9.12e5T^{2} \) |
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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
−10.26454710321227613579509255114, −9.633305593420359041865693077509, −8.373355984588003488891491354113, −7.74957954110986920824125300476, −6.51489857655699754147568154162, −5.73215814991406824817185981115, −4.66091294546231502271803440339, −3.49120062546264648013116712939, −2.60475633422256225250849077622, −1.24556100229978945553509734081,
1.24556100229978945553509734081, 2.60475633422256225250849077622, 3.49120062546264648013116712939, 4.66091294546231502271803440339, 5.73215814991406824817185981115, 6.51489857655699754147568154162, 7.74957954110986920824125300476, 8.373355984588003488891491354113, 9.633305593420359041865693077509, 10.26454710321227613579509255114
