| L(s) = 1 | + 4.61·2-s − 26.3·3-s − 10.7·4-s − 121.·6-s − 179.·7-s − 197.·8-s + 453.·9-s − 605.·11-s + 283.·12-s − 90.2·13-s − 826.·14-s − 565.·16-s − 986.·17-s + 2.09e3·18-s + 424.·19-s + 4.73e3·21-s − 2.79e3·22-s + 4.05e3·23-s + 5.20e3·24-s − 416.·26-s − 5.55e3·27-s + 1.92e3·28-s − 2.81e3·29-s + 3.67e3·31-s + 3.69e3·32-s + 1.59e4·33-s − 4.54e3·34-s + ⋯ |
| L(s) = 1 | + 0.815·2-s − 1.69·3-s − 0.335·4-s − 1.38·6-s − 1.38·7-s − 1.08·8-s + 1.86·9-s − 1.50·11-s + 0.567·12-s − 0.148·13-s − 1.12·14-s − 0.552·16-s − 0.827·17-s + 1.52·18-s + 0.269·19-s + 2.34·21-s − 1.22·22-s + 1.59·23-s + 1.84·24-s − 0.120·26-s − 1.46·27-s + 0.463·28-s − 0.620·29-s + 0.686·31-s + 0.638·32-s + 2.55·33-s − 0.674·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 43 | \( 1 - 1.84e3T \) |
| good | 2 | \( 1 - 4.61T + 32T^{2} \) |
| 3 | \( 1 + 26.3T + 243T^{2} \) |
| 7 | \( 1 + 179.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 605.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 90.2T + 3.71e5T^{2} \) |
| 17 | \( 1 + 986.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 424.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.05e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.81e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.67e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.30e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.66e4T + 1.15e8T^{2} \) |
| 47 | \( 1 + 5.79e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.92e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.48e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.54e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.25e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.73e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.27e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.12e4T + 8.58e9T^{2} \) |
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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
−9.002470414666910991188706382129, −7.56181273868682417573555840027, −6.59859476462003730346961924788, −6.08049292438442778660177446520, −5.16386873633018390392994181458, −4.81074843845568026222414574364, −3.59112757588956910505593390671, −2.62782706737682569714579378892, −0.67351833745014881379740066975, 0,
0.67351833745014881379740066975, 2.62782706737682569714579378892, 3.59112757588956910505593390671, 4.81074843845568026222414574364, 5.16386873633018390392994181458, 6.08049292438442778660177446520, 6.59859476462003730346961924788, 7.56181273868682417573555840027, 9.002470414666910991188706382129
