| L(s) = 1 | − 1.41i·2-s − 1.73·3-s − 2.00·4-s + (−4.91 − 0.905i)5-s + 2.44i·6-s + (1.91 + 6.73i)7-s + 2.82i·8-s + 2.99·9-s + (−1.28 + 6.95i)10-s + 17.5·11-s + 3.46·12-s − 4.83·13-s + (9.52 − 2.70i)14-s + (8.51 + 1.56i)15-s + 4.00·16-s + 18.0·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577·3-s − 0.500·4-s + (−0.983 − 0.181i)5-s + 0.408i·6-s + (0.273 + 0.961i)7-s + 0.353i·8-s + 0.333·9-s + (−0.128 + 0.695i)10-s + 1.59·11-s + 0.288·12-s − 0.371·13-s + (0.680 − 0.193i)14-s + (0.567 + 0.104i)15-s + 0.250·16-s + 1.06·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0946i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.03193 + 0.0489299i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03193 + 0.0489299i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 + (4.91 + 0.905i)T \) |
| 7 | \( 1 + (-1.91 - 6.73i)T \) |
| good | 11 | \( 1 - 17.5T + 121T^{2} \) |
| 13 | \( 1 + 4.83T + 169T^{2} \) |
| 17 | \( 1 - 18.0T + 289T^{2} \) |
| 19 | \( 1 - 9.13iT - 361T^{2} \) |
| 23 | \( 1 - 3.72iT - 529T^{2} \) |
| 29 | \( 1 - 1.12T + 841T^{2} \) |
| 31 | \( 1 - 57.0iT - 961T^{2} \) |
| 37 | \( 1 - 41.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 11.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 64.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 77.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 77.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 87.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 5.36iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 47.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 58.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 53.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 74.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 28.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 107.T + 9.40e3T^{2} \) |
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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
−12.14172306630477590109982128576, −11.51815048690286162330699030976, −10.39905510217197947206470876307, −9.220848310581383197854516105232, −8.392409681182087313495105583794, −7.08358499850877777674319610435, −5.68167845064091751364525290486, −4.52464922340156559888713601974, −3.33524501758920191457286235803, −1.31905674929647061939535401690,
0.76768804751906687779987661325, 3.78249822620732561430947766359, 4.54514488971990033855671351137, 6.06054388423290273660338776150, 7.14866771565670694740129483933, 7.72167050884867811267224983489, 9.072580049851373834533380410970, 10.19384114325297282896648323027, 11.32924584400868815522197292449, 11.96739167774265480134547200326
