| L(s) = 1 | − 4.18·2-s − 14.4·4-s − 96.8·5-s + 111.·7-s + 194.·8-s + 405.·10-s − 71.4·13-s − 466.·14-s − 350.·16-s − 2.28e3·17-s + 282.·19-s + 1.40e3·20-s − 2.39e3·23-s + 6.26e3·25-s + 299.·26-s − 1.61e3·28-s − 209.·29-s + 8.87e3·31-s − 4.75e3·32-s + 9.54e3·34-s − 1.08e4·35-s − 1.35e3·37-s − 1.18e3·38-s − 1.88e4·40-s + 1.20e4·41-s − 1.10e4·43-s + 1.00e4·46-s + ⋯ |
| L(s) = 1 | − 0.739·2-s − 0.452·4-s − 1.73·5-s + 0.860·7-s + 1.07·8-s + 1.28·10-s − 0.117·13-s − 0.636·14-s − 0.342·16-s − 1.91·17-s + 0.179·19-s + 0.784·20-s − 0.944·23-s + 2.00·25-s + 0.0867·26-s − 0.389·28-s − 0.0463·29-s + 1.65·31-s − 0.821·32-s + 1.41·34-s − 1.49·35-s − 0.162·37-s − 0.133·38-s − 1.86·40-s + 1.12·41-s − 0.912·43-s + 0.698·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 4.18T + 32T^{2} \) |
| 5 | \( 1 + 96.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 111.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 71.4T + 3.71e5T^{2} \) |
| 17 | \( 1 + 2.28e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 282.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.39e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 209.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.35e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.20e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.10e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.49e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.20e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.43e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.56e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.16e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 9.94e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.58e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.62e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.02e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.15e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.60e4T + 8.58e9T^{2} \) |
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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
−8.416072817049720833141839426300, −8.178276349047612112031119365154, −7.43044648793483697463997337808, −6.50314441119101596203586088477, −4.78908502671684690213259116834, −4.52145985562218950790569863901, −3.58942125267229981095998913227, −2.10029808407359156218781842632, −0.806013632319958389905898764883, 0,
0.806013632319958389905898764883, 2.10029808407359156218781842632, 3.58942125267229981095998913227, 4.52145985562218950790569863901, 4.78908502671684690213259116834, 6.50314441119101596203586088477, 7.43044648793483697463997337808, 8.178276349047612112031119365154, 8.416072817049720833141839426300
