L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 75.4·5-s − 36·6-s + 64·8-s + 81·9-s − 301.·10-s − 149.·11-s − 144·12-s + 349.·13-s + 679.·15-s + 256·16-s − 1.14e3·17-s + 324·18-s − 2.79e3·19-s − 1.20e3·20-s − 597.·22-s + 1.81e3·23-s − 576·24-s + 2.57e3·25-s + 1.39e3·26-s − 729·27-s − 759.·29-s + 2.71e3·30-s + 9.03e3·31-s + 1.02e3·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.35·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.954·10-s − 0.372·11-s − 0.288·12-s + 0.573·13-s + 0.779·15-s + 0.250·16-s − 0.964·17-s + 0.235·18-s − 1.77·19-s − 0.675·20-s − 0.263·22-s + 0.715·23-s − 0.204·24-s + 0.823·25-s + 0.405·26-s − 0.192·27-s − 0.167·29-s + 0.551·30-s + 1.68·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.650518134\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.650518134\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 + 9T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 75.4T + 3.12e3T^{2} \) |
| 11 | \( 1 + 149.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 349.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.14e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.79e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.81e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 759.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.79e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.64e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.21e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.45e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.35e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.63e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.85e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.23e4T + 8.58e9T^{2} \) |
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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
−11.08979800950461414429127788226, −10.52083749087863782083296939466, −8.835175136012454143515272238903, −7.892436591561497192771935874752, −6.86741096192697068231315386162, −5.96893771827147647389312265714, −4.54192417118133853200706088505, −4.04388929474996239999347815924, −2.54027532383503418490992894949, −0.65932737831927643651496013680,
0.65932737831927643651496013680, 2.54027532383503418490992894949, 4.04388929474996239999347815924, 4.54192417118133853200706088505, 5.96893771827147647389312265714, 6.86741096192697068231315386162, 7.892436591561497192771935874752, 8.835175136012454143515272238903, 10.52083749087863782083296939466, 11.08979800950461414429127788226