| L(s) = 1 | + (−5.46 + 1.45i)2-s − 10.1i·3-s + (27.7 − 15.9i)4-s − 11.8i·5-s + (14.7 + 55.4i)6-s + 105.·7-s + (−128. + 127. i)8-s + 139.·9-s + (17.1 + 64.5i)10-s + 121i·11-s + (−161. − 281. i)12-s + 322. i·13-s + (−577. + 153. i)14-s − 119.·15-s + (517. − 883. i)16-s − 462.·17-s + ⋯ |
| L(s) = 1 | + (−0.966 + 0.257i)2-s − 0.651i·3-s + (0.867 − 0.497i)4-s − 0.211i·5-s + (0.167 + 0.629i)6-s + 0.814·7-s + (−0.710 + 0.703i)8-s + 0.575·9-s + (0.0543 + 0.204i)10-s + 0.301i·11-s + (−0.323 − 0.564i)12-s + 0.528i·13-s + (−0.786 + 0.209i)14-s − 0.137·15-s + (0.505 − 0.863i)16-s − 0.388·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.23602 - 0.515192i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23602 - 0.515192i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.46 - 1.45i)T \) |
| 11 | \( 1 - 121iT \) |
| good | 3 | \( 1 + 10.1iT - 243T^{2} \) |
| 5 | \( 1 + 11.8iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 105.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 322. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 462.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 620. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.39e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 8.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.60e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.23e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.64e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.03e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.07e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 9.97e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 1.48e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.94e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 4.18e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.88e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.32e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 7.50e4T + 8.58e9T^{2} \) |
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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
−12.95649086053493140645440211729, −11.81187241319051515783707792936, −10.85543975394388736816752270658, −9.577226086856040880834594624970, −8.460921890782666028757338950828, −7.42366945684451196084121712990, −6.49773741955865892750601968868, −4.77618412088572304655608313092, −2.17227474946574946370650743844, −0.921954553013698945380058789684,
1.25599438469824086358066559222, 3.10429500995047035209231738838, 4.78846951775582263949397554456, 6.64623883023300417673507638300, 7.930997124115164793847789297722, 8.948747923026940997247234124275, 10.18584518691277909085665391641, 10.82684719748298489670676648071, 11.89824726085813039548580104111, 13.17036607216202049287142241254
