| L(s) = 1 | + (2 + 3.46i)2-s + (14.6 + 5.19i)3-s + (−7.99 + 13.8i)4-s + (1.19 − 2.07i)5-s + (11.3 + 61.3i)6-s + (25.8 + 44.7i)7-s − 63.9·8-s + (189 + 152. i)9-s + 9.57·10-s + (−335. − 581. i)11-s + (−189. + 162. i)12-s + (423. − 733. i)13-s + (−103. + 179. i)14-s + (28.3 − 24.2i)15-s + (−128 − 221. i)16-s − 1.13e3·17-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (0.942 + 0.333i)3-s + (−0.249 + 0.433i)4-s + (0.0214 − 0.0370i)5-s + (0.129 + 0.695i)6-s + (0.199 + 0.345i)7-s − 0.353·8-s + (0.777 + 0.628i)9-s + 0.0302·10-s + (−0.835 − 1.44i)11-s + (−0.380 + 0.324i)12-s + (0.694 − 1.20i)13-s + (−0.140 + 0.244i)14-s + (0.0325 − 0.0278i)15-s + (−0.125 − 0.216i)16-s − 0.952·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.70987 + 1.00834i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70987 + 1.00834i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 - 3.46i)T \) |
| 3 | \( 1 + (-14.6 - 5.19i)T \) |
| good | 5 | \( 1 + (-1.19 + 2.07i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-25.8 - 44.7i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (335. + 581. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-423. + 733. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + 1.13e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.00e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (1.20e3 - 2.07e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (1.88e3 + 3.26e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (2.01e3 - 3.49e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 1.45e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-3.94e3 + 6.82e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.01e4 - 1.76e4i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-8.06e3 - 1.39e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + 5.54e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-3.20e3 + 5.55e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (4.44e3 + 7.69e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.14e4 + 3.71e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 6.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-1.46e3 - 2.53e3i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-2.37e4 - 4.11e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 - 8.47e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (5.58e4 + 9.67e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−17.86105684230720304584628448099, −16.01295061274648738798662263329, −15.42430084666835328961304996437, −13.91870354183291055219471895792, −13.07345333670908913539755920854, −10.86555498261630164020485736686, −8.915779079601797956667019923252, −7.82096121188874813614081769066, −5.50062149911938647193123872670, −3.27599549842234091842542519150,
2.08851336967836819239291254926, 4.28784310012076385842983112758, 7.08629286298606661071889121808, 8.931740363194924856166255454556, 10.42152559117605619823078976369, 12.23125130720910284910479610614, 13.44021279852509992181761484535, 14.46939684041557378488797018692, 15.78273594128648547359147261726, 17.91195569327184199555978887773
